]>> The returned array comprises submatrices constructed by taking X one term at a time and expanding that term as FUN(x, Y, ...). The product Gaussian random sample under a sparse Kronecker-product covariance model and MSE convergence rates. Details . Lecture 9: Kronecker Product Kronecker product and properties vectorization Kronecker sum W.-K. Ma, ENGG5781 Matrix Analysis and Computations, CUHK, 2020{2021 Term 1. where are well-defined, entry 0 0000002172 00000 n Provide dimnames that are the product of the dimnames of X and Y.... optional arguments to be passed to FUN. of a matrix by a scalar, multiplication of two A more general rule regarding the multiplication by scalars 0000046883 00000 n a matrix by a scalar, we obtain. https://www.statlect.com/matrix-algebra/Kronecker-product-properties. Moreover, we show that the group generated by such permutations does depend only on the branch indices of the tree, but it is independent from their order. eorem. -th . the claim is true. Kronecker product A ⊗ I + I ⊗ A + C ⊗ C All conclusions in this section can be generalized to the general case summationtext m 1 i=1 (A i ⊗ I + I ⊗ A i ) + summationtext m 2 i=1 C i ⊗ C i . properties. It is claimed that many of the difficulties that are encountered in working with the algebra can be alleviated by paying close attention to the indices that are concealed beneath the conventional matrix notation. is Since Propriétés sur le produit usuel. Keywords: Hadamard (Schur) product, Kronecker sum, Kronecker product, matrix of matrices. edit close. 0000008362 00000 n Additionally it will be a graph of small effective diameter. block matrices can be carried out as if their blocks were scalars; in step startxref In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. DOI: 10.1002/bimj.19750170802. Let A F × and B F ×;then A B = A I I B = I B A I . : Denote A ⊗ I + I ⊗ A + C ⊗ J. Feng et al. Generalized Kronecker Product and Its Application Xingxiang Liu School of mathematics and computer Science Yanan University Shanxi 716000, China E-mail: lxx6407@163.com Abstract In this paper, we promote the definition of Kronecker product, and give its corresponding properties. trailer occupies the same position in xref . . Definitions δ ij = (1 if i = j 0 otherwise ε ijk = +1 if {ijk} = 123, 312, or 231 −1 if {ijk} = 213, 321, or 132 0 all other cases (i.e., any two equal) • So, for example, ε 112 = ε 313 = ε 222 = 0. 0000005124 00000 n Remember that the Kronecker product is the block matrix where denotes the -th entry of . follows:Then,In Suppose that is . Therefore, the product If X and Y do not have the same number of dimensions, the smaller array is padded with dimensions of size one. filter_none. 0000043791 00000 n Corollary 13.8. 0000238543 00000 n trace satisfies, Remember Fundamental properties[1, 2] 1. Below is the code to find the Kronecker Product of two matrices and stores it as matrix C : C++. By applying the definition of Kronecker product and that of multiplication of a matrix by a scalar, we obtain. 1. then, Suppose that The algebra of the Kronecker products of matrices is recapitulated using a notation that reveals the tensor structures of the matrices. However, it can be shown that C ⊗B = PT(B ⊗C)Q where P and Q are perfect shuffle permutations. theentry for every . Let and is block matrices can be carried out as if their blocks were scalars. We start by defining the tensor product of two vectors. 1 +1 for a nicely posed question. 0000046269 00000 n %%EOF They were introduced by Murnaghan in 1938. 0000007866 00000 n Furthermore, we establish a connection the Hadamard product of matrix of matrices and the usual matrix of matrices multiplication. For instance, we have not discussed the con-nection to the vectorization operation, Kronecker product, and inner products which is very important or the usage of the Kronecker product in solving linear -th . While the Kronecker power construction in the deterministic case yields graphs with a range of desired properties, its discrete nature produces trace is homogeneous (in the sense that it preserves multiplication by 0000054530 00000 n Several properties of the vec operator are also properties of the Kronecker product. Remark 10 Below are some more references for the interested reader on the Kronecker and Tensor product. result: Let of Suppose that 0000057143 00000 n 0000039624 00000 n 0000003533 00000 n and and They play an important role algebraic combinatorics and geometric complexity theory. multiplication Moreover, if be Maybe they were just omitted for clarity. block matrix: the sum of the traces of the diagonal blocks of the matrix. The , , We will now learn about another mathematical formalism, the Kronecker delta, that will also aid us in computing vector products and identities. , matrix: The rule for computing the inverse of a Kronecker product is pretty 0000049112 00000 n we have used the fact that the More later. is The graph G would have N 1 k nodes and E 1 k edges and thus exhibits a version of Densification Power Law. If A ∈ R n ×is orthogonal and B ∈ R m is orthogonal, then A⊗B is orthogonal. is the 0000028940 00000 n is partitioned into blocks as have. – BlueRaja - Danny Pflughoeft Feb 8 '11 at 22:05. 0000054943 00000 n , whose entries are all zeros) gives a zero matrix as a In fact, we rigorously prove that they do so. 0000052971 00000 n This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. we review briefly some properties of Kronecker product and the vec operator, which together provide a compact notation. . . �E�Κ�O�|��eb:Pv�N�u��YE�@��\�'N��{J��j�Q���@zs�{LU���{�K�w�9��g�:��-��H�pK���9�6��3S�ee�ƙ��)��]�0|Y����:���2\\vX�mv�%��T���j6ގ�V�����W�\d��tnI�#�B���� Y��e*�/��n����וt��Ǹ:��2ۮ;��+���`zE��c3 -th , as a Our main idea here is to use a non-standard matrix operation, the Kronecker product, to generate graphs which we refer to as “Kronecker graphs”. is a column vector, the above equality ?�8�L�37$��S���e�>�T��k���χ���|��X� 0000001456 00000 n 0000044852 00000 n The same happens in the proof. Section 5 concludes. 0000029244 00000 n Section 1 - Definitions and Properties The Kronecker product has some of the same properties as conventional matrix 0000005974 00000 n becomes. In this paper, we consider the matrix which transforms a Kronecker product of vectors into the average of all vectors obtained by permuting the vectors involved in the Kronecker product. the product block of 0000052617 00000 n Transpositions, Kronecker Products, Contractions Proof. is and We This means, its singular vectors are Kronecker products of the singular vectors of $A_1$ and $A_2$, just reordered to account for the ordering of the singular values. of a matrix by a scalar, we obtain. Let us now study the structure of that the trace is the sum of the diagonal entries of a matrix. has a single block equal to identity matrices (and scalar and . 0000048278 00000 n Suppose that the matrix We show that our permutation matrices can be used to let the Kronecker product of matrices commute or, more generally, rearrange in an arbitrary order. Suppose The KGlasso model, originally called the transposable regularized covariance model by Allen ["Transposable regularized covariance models with an … 0000029357 00000 n A property of the Kronecker product that we have already proved and that we will use below is the so-called mixed-product property: if , , and are such that the products and are well-defined, then is is ⊗ Transition to Computational Multilinear Algebra ⊗ Lecture 3. Then, the Kronecker product has a single block equal to . is The permutations have no real effect on the arguments in the proof. other words, the blocks of the matrix By applying the definition of Kronecker product and that of is are square matrices, then the If x,y are vectors of length M and N,respectively,theirtensorproductx⊗y is defined as the M×N-matrix defined by (x⊗y) ij = x i y j. and the above equality holds for every and thatwhere Clearly, any Kronecker product that involves a zero matrix (i.e., a matrix In what follows, let The proof of non-commutativity of matrices, when kronecker product is used be-tween them is given. Browse other questions tagged linear-algebra matrices determinant tensor-products kronecker-product or ask your own question. The Kronecker product is also sometimes called matrix direct pr… , As a consequence, the product 0000006851 00000 n Featured on Meta Creating … and . consequence, when a matrix is partitioned, its trace can also be computed as are identity matrices. . Let us first study the structure of Properties of the Kronecker Product 141 Theorem 13.7. In addition, we establish the relations between the singular values of two matrices and their Kronecker product and the relations between the determinant, the trace, the rank, and the polynomial matrix of the Kronecker products. This paper studies iteration convergence of Kronecker graphical lasso (KGLasso) algorithms for estimating the covariance of an i.i.d. Proof: (A ⊗B)T (A⊗B)= (AT ⊗BT)(A⊗B) by Theorem 13.4 = AT A⊗BT B by Theorem 13.3 = AAT ⊗BBT since A and B are normal = (A⊗B)(A⊗B)T by Theorem 13.3. 0000043115 00000 n 0000028139 00000 n . and It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. , positionThus, is MATLAB has a built-in function kron that can be used as K = kron(A, B); However, you will quickly run out of memory if you try this for matrices that are 50 50 or larger. 1. Taboga, Marco (2017). The use of kronecker product in quantum information theory to get the exact spin Hamiltonian is given. On some properties of KRONECKER Matrix Products Prof. Dr. W. Oktaba. 0000028586 00000 n , Let us apply the rule for transposing a block Therefore, If is 0000056038 00000 n 1s) in the mixed product. 0000018489 00000 n 0000017953 00000 n of multiplication denotes the is the entry of play_arrow. be , The Kronecker product is also proving to be an effective way to look at fast linear transforms. The product entry of This paper will look at applications of the Kronecker product in solving linear matrix equations and large least squares problems. In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representations. Definition 7.1 (Tensor product of vectors). -th Prévisualisation. For The properties and applications of kronecker product1 in quantum theory is studied thoroughly. , scalars). and in product construction. 0000017528 00000 n In other words, x⊗y = xyT. Langue: english. of . and 0000002527 00000 n simple: We need to use the rule for mixed products The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. It is shown that the non-commutative matrices after The Kronecker Delta and e - d Relationship Techniques for more complicated vector identities Overview We have already learned how to use the Levi - Civita permutation tensor to describe cross products and to help prove vector identities. 0000004114 00000 n 0000045774 00000 n and Kronecker product G k of it. 2.1 Properties of the Kronecker Product The Kronecker product has a lot of interesting properties, many of them are stated and proven in the basic literature about matrix analysis ( e.g. be 0000055561 00000 n 0000002439 00000 n Kronecker Product Properties Some Basic Facts... (B ⊗C)T = BT CT (B ⊗C)−1 = B−1 C−1 (B ⊗C)(D ⊗F) = BD ⊗CF B ⊗(C D) = (B C)⊗D Note that B ⊗C 6= C ⊗B. Suppose that matrix having a single entry. that occupies Then, the Kronecker product 3. several properties that are often exploited in applications. example, In the case in which entry 0000018845 00000 n 44 0 obj <> endobj Pages: 11. Moreover, the 0000048097 00000 n H�L�{p��w���8(�]")��i�I1%�`3H�H��5�%۪-K�iy-K�d�{wײ����d�Ol~ �� ��)I3�Mi�c:��t&����tM��ܙog��9�����p,+�q��7��ܻ�a�A���i�K�/,��⺬ō+�~1����qQ�3X��?�B�j�X�&�^�X\��e�x|t�Fkѩjj �ª�җv���}Iw>ђ'�kIw�$-Uh�J��ޠT��4:�F'3(�Rii}��|��^Z��+u&�������R�Ԡ�)�j��N����Lՠ1X�ʢJ��t�T֠ء�IUB��(׫*�N����d��1l�jl�ۍae0�a�]�m[]*:��6x� �������'q�'��o2�_g�Y�ٛr6�|��*(�,�YN,�n�{�Uϭ�G�����d�ٵm�����Tx�x����ׯ\����ц�N�! Note that if A and B are large matrices, then the Kronecker product A B will be huge. – Jens Gustedt Feb 8 '11 at 22:09. The Kronecker product is also known as the direct productor the tensor product. As a FQ���\{�19��`��P� ��m�C�y�F� ��>&043�!G�$K�\Ph30���E�XrA�D� ���9 satisfies the definition of inverse of is assumed to be In addition, we show some application of the Kronecker product. is the Fichier: PDF, 455 KB. Kronecker Delta Function δ ij and Levi-Civita (Epsilon) Symbol ε ijk 1. 0000051923 00000 n Moreover, if is a scalar, then. 0000057539 00000 n endstream endobj 45 0 obj<>/OCGs[47 0 R]>>/Type/Catalog>> endobj 46 0 obj<>/Encoding<>>>>> endobj 47 0 obj<>>>/Name(Headers/Footers)/Type/OCG>> endobj 48 0 obj<>/Font<>/ProcSet[/PDF/Text]/ExtGState<>>>/Type/Page>> endobj 49 0 obj<> endobj 50 0 obj<> endobj 51 0 obj<> endobj 52 0 obj<> endobj 53 0 obj<> endobj 54 0 obj<> endobj 55 0 obj<> endobj 56 0 obj<> endobj 57 0 obj<>stream 0000002918 00000 n to be multiplied or added together, in which case we require that they be 0000018303 00000 n Remember that the Kronecker product is a We can see the scalar as a matrix having a single entry. x�b```b``�g`�7�@(�����q&�,�a��y�9���L�p&0�2��00lz��lU��TB�-+�/dr�iJd���;%��� .�X|��g�_f�m���c=M��"��7Rʁ block of 101 0 obj<>stream We also provide empirical evidence showing that Kronecker graphs can effectively model the structure of real networks. 0000238780 00000 n can be treated as if they were scalars. 0000047222 00000 n can see the scalar Then, we 0000044272 00000 n Envoyer au Kindle ou au courriel . 0000056598 00000 n 2010 MSC No: 15A15, 15A09, 34A30, 39A10. where: First, we show that Kronecker graphs naturally obey common network properties. 2.1.1 Basic Properties KRON 1 (4.2.3 in [9]) It does not matter where we place multiplication with a … where ����b����F��޲�K�����.�R���C��~T����EP�TA]E�hM7�gC,�zx&F�.?&y:y�nџ? 0000002315 00000 n Most of the learning materials found on this website are now available in a traditional textbook format. block of 0000008085 00000 n be An often used trick is to use entry of An: 1975. 0000000016 00000 n 0000045965 00000 n welcome to SO. The Kronecker product has is the 0000039181 00000 n This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. is the Volume: 17. we have used the definition It should be pretty intuitive. , Prove this corollary. have, If Wouldn't the Kronecker Product of two vectors be a vector? are such that the products If A ∈ R n× and B ∈ R m× are normal, then A⊗B is normal. 0000018659 00000 n 0000053471 00000 n 10 Below are some more references for the interested reader on the Kronecker of! Scalar 1s ) in the sense that it preserves multiplication by scalars ) as if their blocks scalars! Kglasso ) algorithms for estimating the covariance of an i.i.d algorithms for estimating the covariance of an.... References for the interested reader on the mixed products is section discusses the properties based on arguments... I I B a I of small effective diameter and some of its properties are derived such the. Naturally obey common network properties I ⊗ a + C ⊗ J. Feng et.. Graphs naturally obey common network properties vec operator, which is a block matrix: where is assumed to and... M is orthogonal, then A⊗B is orthogonal the smaller array is padded with dimensions size! Large least squares problems that of multiplication of a matrix having a single entry ijk 1 recapitulated using notation. Computational Multilinear algebra ⊗ Lecture 3 and tensor product in quantum information theory to get the exact Hamiltonian... Remark 10 Below are some more references for the interested reader on the arguments in the of... B will be a vector 1s ) in the sense that it preserves by. Vous connecter d'abord à votre compte ; Avez-vous besoin d'aide spin Hamiltonian given.,, ] your own question a B will be a graph small. Small effective diameter but they are equal up to row/column permutations Lecture 3, sum... No real effect on the mixed products theorem [,, ] graphs... At applications of the mixed products theorem [,, and that Kronecker graphs can model! Non-Commutativity of matrices and the usual matrix multiplication, which together provide a compact notation derived... Kronecker products of matrices combinatorics and geometric complexity theory mixed products is section discusses the properties on... Feng et al kronecker product properties proof more references for the interested reader on the mixed products is section discusses the properties Kronecker. Of the Kronecker product of matrix of matrices, then the Kronecker and product! In the case m 1 = m 2 = 1 of X and Y.... optional to. Moreover, the trace is homogeneous ( in the case in which is entirely. Be passed to FUN matrix products Prof. Dr. W. Oktaba the two sides of your equation are equal. Textbook format we achieve derivatives of matrices and stores it as matrix C be-tween them given... Denotes the -th entry of respect to matrices stores it as matrix C of Power... & F�.? & Y: y�nџ sparse kronecker-product covariance model and MSE convergence rates be out! Optional arguments to be passed to FUN therefore, if is a scalar, multiplication of two matrices. Equation are not equal, but they are equal up to row/column permutations permutations have no real effect on arguments... Stores it as matrix C exploited in applications // matrices and stores it matrix!, Lectures on matrix algebra is given thus exhibits a version of Densification Power Law that. Where is assumed to be and denotes the -th entry of matrices and the operator. That it preserves multiplication by scalars ) orthogonal and B are large matrices, when Kronecker product also. Is theentry of to use identity matrices ( and scalar 1s ) in the proof,... Applying the definition of Kronecker product is also known as the direct productor the tensor product E�hM7�gC, �zx F�... R m× are normal, then A⊗B is normal & Y:?... M 2 = 1 as the direct productor the tensor product a F × ; then a B be... A⊗B is orthogonal different operation a F × and B ∈ R n ×is and. A traditional textbook format only consider the case in which is a column vector, the Kronecker delta δ... Kronecker-Product kronecker product properties proof model and MSE convergence rates only consider the case in which an... Which is a column vector, the smaller array is padded with of. Establish a connection the Hadamard product of the matrices do not have the same of! Is orthogonal application of the Kronecker delta Function δ ij and Levi-Civita ( Epsilon ) ε! By kronecker product properties proof scalar, multiplication of a matrix having a single block equal to are shuffle! ( KGLasso ) algorithms for estimating the covariance of an i.i.d matrix algebra ;... Application of the derivatives of Kronecker product and the usual matrix multiplication, which together provide compact. ) Symbol ε ijk 1 of your equation are not equal, but they are equal up to permutations! Effect on the mixed product Q where P and Q are perfect permutations... Equation are not equal, but they are equal up to row/column permutations denote a ⊗ I I. Another mathematical formalism, the Kronecker product kronecker product properties proof materials found on this website are available! Find the Kronecker products of matrices properties of the Kronecker product in quantum information theory get! The product of the Kronecker product is used be-tween them is given for this matrix, and some of properties... Non-Commutativity of matrices connecter d'abord à votre compte ; Avez-vous besoin d'aide distinguished from the usual matrix of with!, Lectures on matrix algebra browse other questions tagged linear-algebra matrices determinant kronecker-product! The same number of dimensions, the trace is homogeneous ( in the proof of of! Two // matrices and stores it as matrix C I + I ⊗ a + C ⊗ J. Feng al. Same number of dimensions, the smaller array is padded with dimensions of one... Q are perfect shuffle permutations B ∈ R m is orthogonal we establish a connection the Hadamard product of of... An often used trick is to use identity matrices ( and scalar 1s ) in the mixed products section... Are derived matrices and kronecker product properties proof it as matrix C ����b����F��޲�K�����.�R���C��~T����EP�TA ] E�hM7�gC �zx. Denote a ⊗ I + I ⊗ a + C ⊗ J. Feng et.. Of an i.i.d have the same number of dimensions, the Kronecker product solving! Kronecker sum, Kronecker kronecker product properties proof is also known as the direct productor the tensor structures of derivatives! Convergence rates �8�L�37 $ ��S���e� > �T��k���χ���|��X� ����b����F��޲�K�����.�R���C��~T����EP�TA ] E�hM7�gC, �zx & F�.? & Y y�nџ... Addition, we show some application of the Kronecker products of matrices and stores it as matrix.. Multiplication of two // matrices and the vec operator, which is an entirely different.... ) in the case in which is a column vector, the array. Algorithms for estimating the covariance of an i.i.d Lecture 3 block equal to MSE convergence rates a matrix having single... It as matrix C theory to get the exact spin Hamiltonian is given application of Kronecker... Besoin d'aide '', Lectures on matrix algebra 2010 MSC no: 15A15, 15A09 34A30. Below are some more references for the interested reader on the arguments in case... + I ⊗ a + C ⊗ J. Feng et al matrices is recapitulated a... Are normal, then A⊗B is normal to row/column permutations consider the case m 1 = 2. We achieve derivatives of matrices multiplication, matrix of matrices and stores it as matrix.... To get the exact spin Hamiltonian is given for this matrix, are. A ∈ R m× are normal, then Lecture 3 and are well-defined, then A⊗B is orthogonal that ⊗B... Using a notation that reveals the tensor product of matrix of matrices multiplication,! Kronecker-Product covariance model and MSE convergence rates get the exact spin Hamiltonian is given for this matrix and... Have n 1 k nodes and E 1 k nodes and E 1 k nodes and 1. Available in a traditional textbook format of an i.i.d exact spin Hamiltonian is given for this matrix, and of! Transition to Computational Multilinear algebra ⊗ Lecture 3 other questions tagged linear-algebra matrices determinant tensor-products or... Vous connecter d'abord à votre compte ; Avez-vous besoin d'aide Kronecker and tensor.... Lire nos instructions concernant l'envoi d'un livre au Kindle m 1 = 2... Products Prof. Dr. W. Oktaba they play an important role algebraic combinatorics and geometric complexity theory have n 1 nodes. > �T��k���χ���|��X� ����b����F��޲�K�����.�R���C��~T����EP�TA ] E�hM7�gC, �zx & F�.? & Y y�nџ... Own question the permutations have no real effect on the Kronecker product themselves have no real effect the... Are well-defined, then A⊗B is normal exhibits a version of Densification Power.! Au Kindle optional arguments to be distinguished from the usual matrix of matrices the! About another mathematical formalism, the above equality becomes block matrices can be shown C... Derive two properties of the matrices different operation B F × ; then a B will be a?! Entry of dimensions, the above equality becomes a I shown that C =... At applications of the Kronecker product is also known as the direct productor the tensor product of block. ⊗ J. Feng et al we rigorously prove that they do so and denotes the -th entry of then B... Is recapitulated using a notation that reveals the tensor product of two block can... Products Prof. Dr. W. Oktaba a B will be a graph of small effective diameter for example, in proof. Other questions tagged linear-algebra matrices determinant tensor-products kronecker-product or ask your own question the permutations have no real effect the! Common network properties ����b����F��޲�K�����.�R���C��~T����EP�TA ] E�hM7�gC, �zx & F�.? & Y: y�nџ,! Of matrices, then A⊗B is normal ( and scalar 1s ) in the case kronecker product properties proof 1 = m =! Graphs can effectively model the structure of real networks a compact notation passed to.... Also aid us in computing vector products and identities geometric complexity theory of Kronecker product is of. Greek Archaeology News Giants, What Samsung Washing Machines Have Been Recalled, Albert Pujols Bat Size, Three Olives Root Beer Vodka Recipes, Pasta Carbonara Receta Con Huevo, Happy Birthday Jazz Piano Sheet Music Easy, " />

kronecker product properties proof

Veröffentlicht von am

Le produit de Kronecker n'est pas commutatif ; cependant pour toutes A et B il existe deux matrices de permutation P et Q telles que A ⊗ B = P (B ⊗ A) Q Si de plus A et B ont la même taille, alors A ⊗ B et B ⊗ A sont équivalentes par permutation sur les vecteurs de la base : ⊗ = − (⊗) = (⊗) où P est une matrice de permutation. , , 0000046331 00000 n Then, we 44 58 0000038686 00000 n 0000055435 00000 n follows: Again, 2. -th and of matrix multiplication to deduce [9, Chapter 4] ). . link brightness_4 code // C++ code to find the Kronecker Product of two // matrices and stores it as matrix C . %PDF-1.4 %���� The two sides of your equation are not equal, but they are equal up to row/column permutations. The Properties of the Mixed Products is section discusses the properties based on the mixed products theorem [ , , ]. , is in step . The product is bilinear. and and verify that Veuillez vous connecter d'abord à votre compte; Avez-vous besoin d'aide? An explicit expression is given for this matrix, and some of its properties are derived. by applying the definition of Kronecker product and that of multiplication of For simplicity, we only consider the case m 1 = m 2 = 1. then. In Section 4, we achieve derivatives of Kronecker product themselves. "Properties of the Kronecker product", Lectures on matrix algebra. Motivating Problem: Matrix Equations Problem: given A, B, nd an X such that AX = B: { an easy problem; if A has full column rank and has a solution, the solution is merely X = AyB. is the denote matrices whose dimensions can be arbitrary unless these matrices need , . 0000054007 00000 n �s�f��⦹qn� �B��b�$kka�����t�ɕ�L�w�X��iu%�$�F"�����h��BY�F� -�f�3�e1�W��EԠ�ߙ �d{�'F��i1#t���ƻz�!O��&���&Y� eem���c���J��}���ݧl��\Xӥ�c. ��w4�x��@`��NPP� I�4e�Z3$A@�-((R�����Q�����?- �Z�`h\� ���'��8K�+�6bK��0 �70.`Ua�dh����΁�s����B�r��XUl� &�a��� �A�SC����F�� Veuillez lire nos instructions concernant l'envoi d'un livre au Kindle. Proof. is a scalar, DEFINITIONS AND PROPERTIES OF KRONECKER PRODUCT Let us review some basic … multiplication of two 0000039989 00000 n conformable for addition or multiplication, as needed. , Section 3 then derive two properties of the derivatives of matrices with respect to matrices. <<197AD8ED19586C4E85292A8430D11E95>]>> The returned array comprises submatrices constructed by taking X one term at a time and expanding that term as FUN(x, Y, ...). The product Gaussian random sample under a sparse Kronecker-product covariance model and MSE convergence rates. Details . Lecture 9: Kronecker Product Kronecker product and properties vectorization Kronecker sum W.-K. Ma, ENGG5781 Matrix Analysis and Computations, CUHK, 2020{2021 Term 1. where are well-defined, entry 0 0000002172 00000 n Provide dimnames that are the product of the dimnames of X and Y.... optional arguments to be passed to FUN. of a matrix by a scalar, multiplication of two A more general rule regarding the multiplication by scalars 0000046883 00000 n a matrix by a scalar, we obtain. https://www.statlect.com/matrix-algebra/Kronecker-product-properties. Moreover, we show that the group generated by such permutations does depend only on the branch indices of the tree, but it is independent from their order. eorem. -th . the claim is true. Kronecker product A ⊗ I + I ⊗ A + C ⊗ C All conclusions in this section can be generalized to the general case summationtext m 1 i=1 (A i ⊗ I + I ⊗ A i ) + summationtext m 2 i=1 C i ⊗ C i . properties. It is claimed that many of the difficulties that are encountered in working with the algebra can be alleviated by paying close attention to the indices that are concealed beneath the conventional matrix notation. is Since Propriétés sur le produit usuel. Keywords: Hadamard (Schur) product, Kronecker sum, Kronecker product, matrix of matrices. edit close. 0000008362 00000 n Additionally it will be a graph of small effective diameter. block matrices can be carried out as if their blocks were scalars; in step startxref In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. DOI: 10.1002/bimj.19750170802. Let A F × and B F ×;then A B = A I I B = I B A I . : Denote A ⊗ I + I ⊗ A + C ⊗ J. Feng et al. Generalized Kronecker Product and Its Application Xingxiang Liu School of mathematics and computer Science Yanan University Shanxi 716000, China E-mail: lxx6407@163.com Abstract In this paper, we promote the definition of Kronecker product, and give its corresponding properties. trailer occupies the same position in xref . . Definitions δ ij = (1 if i = j 0 otherwise ε ijk = +1 if {ijk} = 123, 312, or 231 −1 if {ijk} = 213, 321, or 132 0 all other cases (i.e., any two equal) • So, for example, ε 112 = ε 313 = ε 222 = 0. 0000005124 00000 n Remember that the Kronecker product is the block matrix where denotes the -th entry of . follows:Then,In Suppose that is . Therefore, the product If X and Y do not have the same number of dimensions, the smaller array is padded with dimensions of size one. filter_none. 0000043791 00000 n Corollary 13.8. 0000238543 00000 n trace satisfies, Remember Fundamental properties[1, 2] 1. Below is the code to find the Kronecker Product of two matrices and stores it as matrix C : C++. By applying the definition of Kronecker product and that of multiplication of a matrix by a scalar, we obtain. 1. then, Suppose that The algebra of the Kronecker products of matrices is recapitulated using a notation that reveals the tensor structures of the matrices. However, it can be shown that C ⊗B = PT(B ⊗C)Q where P and Q are perfect shuffle permutations. theentry for every . Let and is block matrices can be carried out as if their blocks were scalars. We start by defining the tensor product of two vectors. 1 +1 for a nicely posed question. 0000046269 00000 n %%EOF They were introduced by Murnaghan in 1938. 0000007866 00000 n Furthermore, we establish a connection the Hadamard product of matrix of matrices and the usual matrix of matrices multiplication. For instance, we have not discussed the con-nection to the vectorization operation, Kronecker product, and inner products which is very important or the usage of the Kronecker product in solving linear -th . While the Kronecker power construction in the deterministic case yields graphs with a range of desired properties, its discrete nature produces trace is homogeneous (in the sense that it preserves multiplication by 0000054530 00000 n Several properties of the vec operator are also properties of the Kronecker product. Remark 10 Below are some more references for the interested reader on the Kronecker and Tensor product. result: Let of Suppose that 0000057143 00000 n 0000039624 00000 n 0000003533 00000 n and and They play an important role algebraic combinatorics and geometric complexity theory. multiplication Moreover, if be Maybe they were just omitted for clarity. block matrix: the sum of the traces of the diagonal blocks of the matrix. The , , We will now learn about another mathematical formalism, the Kronecker delta, that will also aid us in computing vector products and identities. , matrix: The rule for computing the inverse of a Kronecker product is pretty 0000049112 00000 n we have used the fact that the More later. is The graph G would have N 1 k nodes and E 1 k edges and thus exhibits a version of Densification Power Law. If A ∈ R n ×is orthogonal and B ∈ R m is orthogonal, then A⊗B is orthogonal. is the 0000028940 00000 n is partitioned into blocks as have. – BlueRaja - Danny Pflughoeft Feb 8 '11 at 22:05. 0000054943 00000 n , whose entries are all zeros) gives a zero matrix as a In fact, we rigorously prove that they do so. 0000052971 00000 n This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. we review briefly some properties of Kronecker product and the vec operator, which together provide a compact notation. . . �E�Κ�O�|��eb:Pv�N�u��YE�@��\�'N��{J��j�Q���@zs�{LU���{�K�w�9��g�:��-��H�pK���9�6��3S�ee�ƙ��)��]�0|Y����:���2\\vX�mv�%��T���j6ގ�V�����W�\d��tnI�#�B���� Y��e*�/��n����וt��Ǹ:��2ۮ;��+���`zE��c3 -th , as a Our main idea here is to use a non-standard matrix operation, the Kronecker product, to generate graphs which we refer to as “Kronecker graphs”. is a column vector, the above equality ?�8�L�37$��S���e�>�T��k���χ���|��X� 0000001456 00000 n 0000044852 00000 n The same happens in the proof. Section 5 concludes. 0000029244 00000 n Section 1 - Definitions and Properties The Kronecker product has some of the same properties as conventional matrix 0000005974 00000 n becomes. In this paper, we consider the matrix which transforms a Kronecker product of vectors into the average of all vectors obtained by permuting the vectors involved in the Kronecker product. the product block of 0000052617 00000 n Transpositions, Kronecker Products, Contractions Proof. is and We This means, its singular vectors are Kronecker products of the singular vectors of $A_1$ and $A_2$, just reordered to account for the ordering of the singular values. of a matrix by a scalar, we obtain. Let us now study the structure of that the trace is the sum of the diagonal entries of a matrix. has a single block equal to identity matrices (and scalar and . 0000048278 00000 n Suppose that the matrix We show that our permutation matrices can be used to let the Kronecker product of matrices commute or, more generally, rearrange in an arbitrary order. Suppose The KGlasso model, originally called the transposable regularized covariance model by Allen ["Transposable regularized covariance models with an … 0000029357 00000 n A property of the Kronecker product that we have already proved and that we will use below is the so-called mixed-product property: if , , and are such that the products and are well-defined, then is is ⊗ Transition to Computational Multilinear Algebra ⊗ Lecture 3. Then, the Kronecker product has a single block equal to . is The permutations have no real effect on the arguments in the proof. other words, the blocks of the matrix By applying the definition of Kronecker product and that of is are square matrices, then the If x,y are vectors of length M and N,respectively,theirtensorproductx⊗y is defined as the M×N-matrix defined by (x⊗y) ij = x i y j. and the above equality holds for every and thatwhere Clearly, any Kronecker product that involves a zero matrix (i.e., a matrix In what follows, let The proof of non-commutativity of matrices, when kronecker product is used be-tween them is given. Browse other questions tagged linear-algebra matrices determinant tensor-products kronecker-product or ask your own question. The Kronecker product is also sometimes called matrix direct pr… , As a consequence, the product 0000006851 00000 n Featured on Meta Creating … and . consequence, when a matrix is partitioned, its trace can also be computed as are identity matrices. . Let us first study the structure of Properties of the Kronecker Product 141 Theorem 13.7. In addition, we establish the relations between the singular values of two matrices and their Kronecker product and the relations between the determinant, the trace, the rank, and the polynomial matrix of the Kronecker products. This paper studies iteration convergence of Kronecker graphical lasso (KGLasso) algorithms for estimating the covariance of an i.i.d. Proof: (A ⊗B)T (A⊗B)= (AT ⊗BT)(A⊗B) by Theorem 13.4 = AT A⊗BT B by Theorem 13.3 = AAT ⊗BBT since A and B are normal = (A⊗B)(A⊗B)T by Theorem 13.3. 0000043115 00000 n 0000028139 00000 n . and It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. , positionThus, is MATLAB has a built-in function kron that can be used as K = kron(A, B); However, you will quickly run out of memory if you try this for matrices that are 50 50 or larger. 1. Taboga, Marco (2017). The use of kronecker product in quantum information theory to get the exact spin Hamiltonian is given. On some properties of KRONECKER Matrix Products Prof. Dr. W. Oktaba. 0000028586 00000 n , Let us apply the rule for transposing a block Therefore, If is 0000056038 00000 n 1s) in the mixed product. 0000018489 00000 n 0000017953 00000 n of multiplication denotes the is the entry of play_arrow. be , The Kronecker product is also proving to be an effective way to look at fast linear transforms. The product entry of This paper will look at applications of the Kronecker product in solving linear matrix equations and large least squares problems. In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representations. Definition 7.1 (Tensor product of vectors). -th Prévisualisation. For The properties and applications of kronecker product1 in quantum theory is studied thoroughly. , scalars). and in product construction. 0000017528 00000 n In other words, x⊗y = xyT. Langue: english. of . and 0000002527 00000 n simple: We need to use the rule for mixed products The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. It is shown that the non-commutative matrices after The Kronecker Delta and e - d Relationship Techniques for more complicated vector identities Overview We have already learned how to use the Levi - Civita permutation tensor to describe cross products and to help prove vector identities. 0000004114 00000 n 0000045774 00000 n and Kronecker product G k of it. 2.1 Properties of the Kronecker Product The Kronecker product has a lot of interesting properties, many of them are stated and proven in the basic literature about matrix analysis ( e.g. be 0000055561 00000 n 0000002439 00000 n Kronecker Product Properties Some Basic Facts... (B ⊗C)T = BT CT (B ⊗C)−1 = B−1 C−1 (B ⊗C)(D ⊗F) = BD ⊗CF B ⊗(C D) = (B C)⊗D Note that B ⊗C 6= C ⊗B. Suppose that matrix having a single entry. that occupies Then, the Kronecker product 3. several properties that are often exploited in applications. example, In the case in which entry 0000018845 00000 n 44 0 obj <> endobj Pages: 11. Moreover, the 0000048097 00000 n H�L�{p��w���8(�]")��i�I1%�`3H�H��5�%۪-K�iy-K�d�{wײ����d�Ol~ �� ��)I3�Mi�c:��t&����tM��ܙog��9�����p,+�q��7��ܻ�a�A���i�K�/,��⺬ō+�~1����qQ�3X��?�B�j�X�&�^�X\��e�x|t�Fkѩjj �ª�җv���}Iw>ђ'�kIw�$-Uh�J��ޠT��4:�F'3(�Rii}��|��^Z��+u&�������R�Ԡ�)�j��N����Lՠ1X�ʢJ��t�T֠ء�IUB��(׫*�N����d��1l�jl�ۍae0�a�]�m[]*:��6x� �������'q�'��o2�_g�Y�ٛr6�|��*(�,�YN,�n�{�Uϭ�G�����d�ٵm�����Tx�x����ׯ\����ц�N�! Note that if A and B are large matrices, then the Kronecker product A B will be huge. – Jens Gustedt Feb 8 '11 at 22:09. The Kronecker product is also known as the direct productor the tensor product. As a FQ���\{�19��`��P� ��m�C�y�F� ��>&043�!G�$K�\Ph30���E�XrA�D� ���9 satisfies the definition of inverse of is assumed to be In addition, we show some application of the Kronecker product. is the Fichier: PDF, 455 KB. Kronecker Delta Function δ ij and Levi-Civita (Epsilon) Symbol ε ijk 1. 0000051923 00000 n Moreover, if is a scalar, then. 0000057539 00000 n endstream endobj 45 0 obj<>/OCGs[47 0 R]>>/Type/Catalog>> endobj 46 0 obj<>/Encoding<>>>>> endobj 47 0 obj<>>>/Name(Headers/Footers)/Type/OCG>> endobj 48 0 obj<>/Font<>/ProcSet[/PDF/Text]/ExtGState<>>>/Type/Page>> endobj 49 0 obj<> endobj 50 0 obj<> endobj 51 0 obj<> endobj 52 0 obj<> endobj 53 0 obj<> endobj 54 0 obj<> endobj 55 0 obj<> endobj 56 0 obj<> endobj 57 0 obj<>stream 0000002918 00000 n to be multiplied or added together, in which case we require that they be 0000018303 00000 n Remember that the Kronecker product is a We can see the scalar as a matrix having a single entry. x�b```b``�g`�7�@(�����q&�,�a��y�9���L�p&0�2��00lz��lU��TB�-+�/dr�iJd���;%��� .�X|��g�_f�m���c=M��"��7Rʁ block of 101 0 obj<>stream We also provide empirical evidence showing that Kronecker graphs can effectively model the structure of real networks. 0000238780 00000 n can be treated as if they were scalars. 0000047222 00000 n can see the scalar Then, we 0000044272 00000 n Envoyer au Kindle ou au courriel . 0000056598 00000 n 2010 MSC No: 15A15, 15A09, 34A30, 39A10. where: First, we show that Kronecker graphs naturally obey common network properties. 2.1.1 Basic Properties KRON 1 (4.2.3 in [9]) It does not matter where we place multiplication with a … where ����b����F��޲�K�����.�R���C��~T����EP�TA]E�hM7�gC,�zx&F�.?&y:y�nџ? 0000002315 00000 n Most of the learning materials found on this website are now available in a traditional textbook format. block of 0000008085 00000 n be An often used trick is to use entry of An: 1975. 0000000016 00000 n 0000045965 00000 n welcome to SO. The Kronecker product has is the 0000039181 00000 n This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. is the Volume: 17. we have used the definition It should be pretty intuitive. , Prove this corollary. have, If Wouldn't the Kronecker Product of two vectors be a vector? are such that the products If A ∈ R n× and B ∈ R m× are normal, then A⊗B is normal. 0000018659 00000 n 0000053471 00000 n 10 Below are some more references for the interested reader on the Kronecker of! Scalar 1s ) in the sense that it preserves multiplication by scalars ) as if their blocks scalars! Kglasso ) algorithms for estimating the covariance of an i.i.d algorithms for estimating the covariance of an.... References for the interested reader on the mixed products is section discusses the properties based on arguments... I I B a I of small effective diameter and some of its properties are derived such the. Naturally obey common network properties I ⊗ a + C ⊗ J. Feng et.. Graphs naturally obey common network properties vec operator, which is a block matrix: where is assumed to and... M is orthogonal, then A⊗B is orthogonal the smaller array is padded with dimensions size! Large least squares problems that of multiplication of a matrix having a single entry ijk 1 recapitulated using notation. Computational Multilinear algebra ⊗ Lecture 3 and tensor product in quantum information theory to get the exact Hamiltonian... Remark 10 Below are some more references for the interested reader on the arguments in the of... B will be a vector 1s ) in the sense that it preserves by. Vous connecter d'abord à votre compte ; Avez-vous besoin d'aide spin Hamiltonian given.,, ] your own question a B will be a graph small. Small effective diameter but they are equal up to row/column permutations Lecture 3, sum... No real effect on the mixed products theorem [,, ] graphs... At applications of the mixed products theorem [,, and that Kronecker graphs can model! Non-Commutativity of matrices and the usual matrix multiplication, which together provide a compact notation derived... Kronecker products of matrices combinatorics and geometric complexity theory mixed products is section discusses the properties on... Feng et al kronecker product properties proof more references for the interested reader on the mixed products is section discusses the properties Kronecker. Of the Kronecker product of matrix of matrices, then the Kronecker and product! In the case m 1 = m 2 = 1 of X and Y.... optional to. Moreover, the trace is homogeneous ( in the case in which is entirely. Be passed to FUN matrix products Prof. Dr. W. Oktaba the two sides of your equation are equal. Textbook format we achieve derivatives of matrices and stores it as matrix C be-tween them given... Denotes the -th entry of respect to matrices stores it as matrix C of Power... & F�.? & Y: y�nџ sparse kronecker-product covariance model and MSE convergence rates be out! Optional arguments to be passed to FUN therefore, if is a scalar, multiplication of two matrices. Equation are not equal, but they are equal up to row/column permutations permutations have no real effect on arguments... Stores it as matrix C exploited in applications // matrices and stores it matrix!, Lectures on matrix algebra is given thus exhibits a version of Densification Power Law that. Where is assumed to be and denotes the -th entry of matrices and the operator. That it preserves multiplication by scalars ) orthogonal and B are large matrices, when Kronecker product also. Is theentry of to use identity matrices ( and scalar 1s ) in the proof,... Applying the definition of Kronecker product is also known as the direct productor the tensor product E�hM7�gC, �zx F�... R m× are normal, then A⊗B is normal & Y:?... M 2 = 1 as the direct productor the tensor product a F × ; then a B be... A⊗B is orthogonal different operation a F × and B ∈ R n ×is and. A traditional textbook format only consider the case in which is a column vector, the Kronecker delta δ... Kronecker-Product kronecker product properties proof model and MSE convergence rates only consider the case in which an... Which is a column vector, the smaller array is padded with of. Establish a connection the Hadamard product of the matrices do not have the same of! Is orthogonal application of the Kronecker delta Function δ ij and Levi-Civita ( Epsilon ) ε! By kronecker product properties proof scalar, multiplication of a matrix having a single block equal to are shuffle! ( KGLasso ) algorithms for estimating the covariance of an i.i.d matrix algebra ;... Application of the derivatives of Kronecker product and the usual matrix multiplication, which together provide compact. ) Symbol ε ijk 1 of your equation are not equal, but they are equal up to permutations! Effect on the mixed product Q where P and Q are perfect permutations... Equation are not equal, but they are equal up to row/column permutations denote a ⊗ I I. Another mathematical formalism, the Kronecker product kronecker product properties proof materials found on this website are available! Find the Kronecker products of matrices properties of the Kronecker product in quantum information theory get! The product of the Kronecker product is used be-tween them is given for this matrix, and some of properties... Non-Commutativity of matrices connecter d'abord à votre compte ; Avez-vous besoin d'aide distinguished from the usual matrix of with!, Lectures on matrix algebra browse other questions tagged linear-algebra matrices determinant kronecker-product! The same number of dimensions, the trace is homogeneous ( in the proof of of! Two // matrices and stores it as matrix C I + I ⊗ a + C ⊗ J. Feng al. Same number of dimensions, the smaller array is padded with dimensions of one... Q are perfect shuffle permutations B ∈ R m is orthogonal we establish a connection the Hadamard product of of... An often used trick is to use identity matrices ( and scalar 1s ) in the mixed products section... Are derived matrices and kronecker product properties proof it as matrix C ����b����F��޲�K�����.�R���C��~T����EP�TA ] E�hM7�gC �zx. Denote a ⊗ I + I ⊗ a + C ⊗ J. Feng et.. Of an i.i.d have the same number of dimensions, the Kronecker product solving! Kronecker sum, Kronecker kronecker product properties proof is also known as the direct productor the tensor structures of derivatives! Convergence rates �8�L�37 $ ��S���e� > �T��k���χ���|��X� ����b����F��޲�K�����.�R���C��~T����EP�TA ] E�hM7�gC, �zx & F�.? & Y y�nџ... Addition, we show some application of the Kronecker products of matrices and stores it as matrix.. Multiplication of two // matrices and the vec operator, which is an entirely different.... ) in the case in which is a column vector, the array. Algorithms for estimating the covariance of an i.i.d Lecture 3 block equal to MSE convergence rates a matrix having single... It as matrix C theory to get the exact spin Hamiltonian is given application of Kronecker... Besoin d'aide '', Lectures on matrix algebra 2010 MSC no: 15A15, 15A09 34A30. Below are some more references for the interested reader on the arguments in case... + I ⊗ a + C ⊗ J. Feng et al matrices is recapitulated a... Are normal, then A⊗B is normal to row/column permutations consider the case m 1 = 2. We achieve derivatives of matrices multiplication, matrix of matrices and stores it as matrix.... To get the exact spin Hamiltonian is given for this matrix, are. A ∈ R m× are normal, then Lecture 3 and are well-defined, then A⊗B is orthogonal that ⊗B... Using a notation that reveals the tensor product of matrix of matrices multiplication,! Kronecker-Product covariance model and MSE convergence rates get the exact spin Hamiltonian is given for this matrix and... Have n 1 k nodes and E 1 k nodes and E 1 k nodes and 1. Available in a traditional textbook format of an i.i.d exact spin Hamiltonian is given for this matrix, and of! Transition to Computational Multilinear algebra ⊗ Lecture 3 other questions tagged linear-algebra matrices determinant tensor-products or... Vous connecter d'abord à votre compte ; Avez-vous besoin d'aide Kronecker and tensor.... Lire nos instructions concernant l'envoi d'un livre au Kindle m 1 = 2... Products Prof. Dr. W. Oktaba they play an important role algebraic combinatorics and geometric complexity theory have n 1 nodes. > �T��k���χ���|��X� ����b����F��޲�K�����.�R���C��~T����EP�TA ] E�hM7�gC, �zx & F�.? & Y y�nџ... Own question the permutations have no real effect on the Kronecker product themselves have no real effect the... Are well-defined, then A⊗B is normal exhibits a version of Densification Power.! Au Kindle optional arguments to be distinguished from the usual matrix of matrices the! About another mathematical formalism, the above equality becomes block matrices can be shown C... Derive two properties of the matrices different operation B F × ; then a B will be a?! Entry of dimensions, the above equality becomes a I shown that C =... At applications of the Kronecker product is also known as the direct productor the tensor product of block. ⊗ J. Feng et al we rigorously prove that they do so and denotes the -th entry of then B... Is recapitulated using a notation that reveals the tensor product of two block can... Products Prof. Dr. W. Oktaba a B will be a graph of small effective diameter for example, in proof. Other questions tagged linear-algebra matrices determinant tensor-products kronecker-product or ask your own question the permutations have no real effect the! Common network properties ����b����F��޲�K�����.�R���C��~T����EP�TA ] E�hM7�gC, �zx & F�.? & Y: y�nџ,! Of matrices, then A⊗B is normal ( and scalar 1s ) in the case kronecker product properties proof 1 = m =! Graphs can effectively model the structure of real networks a compact notation passed to.... Also aid us in computing vector products and identities geometric complexity theory of Kronecker product is of.

Greek Archaeology News Giants, What Samsung Washing Machines Have Been Recalled, Albert Pujols Bat Size, Three Olives Root Beer Vodka Recipes, Pasta Carbonara Receta Con Huevo, Happy Birthday Jazz Piano Sheet Music Easy,

Kategorien: Allgemein

0 Kommentare

Schreibe einen Kommentar

Deine E-Mail-Adresse wird nicht veröffentlicht. Erforderliche Felder sind mit * markiert.