University:
Massachusetts Institute of Technology
Course:
20.482J | Foundations of Algorithms and Computational Techniques in Systems Biology
Academic year:

2022
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298

Pages:

3
Author:

wezekanaqe

6.581J / 20.482J Foundations of Algorithms and Computational Techniques in Systems Biology Professor Bruce Tidor Professor Jacob K. White Notes on Kronecker Products Department of Mechanical Engineering G.W.C. Whiting School of Engineering The Johns Hopkins University This note is a brief description of the matrix Kronecker product and matrix stack algebraic operators. For a detailed treatment the reader is referred to [1]. 1 The Stack Operator The stack operator maps an n × m matrix into an nm × 1 vector. The stack of the n × m matrix A, denoted AS , is the vector formed by stacking the columns of A into an nm × 1 vector. For example if � A= then its stack form is a c b d � (1) 2×2 ⎡ ⎤ a ⎢ b ⎥ ⎥ AS = ⎢ . ⎣ c ⎦ d 4×1 (2) If C is an n × m matrix comprising m column vectors {c1 , c2 , · · · cm }, where each ci is an n × 1 vector C = [c1 , c2 , · · · , cm ]n×m then ⎡ ⎢ ⎢ CS = ⎢ ⎣ c1 c2 .. . cm 1.1 ⎤ ⎥ ⎥ ⎥ ⎦ . nm×1 Properties of the Stack Operator 1. If v ∈ IRn×1 , a vector, then v S = v. 2. If A ∈ IRm×n , a matrix, and v ∈ IRn×1 , a vector, then the matrix product (Av)S = Av. 1 (3) (4) 3. trace(AB) = ((AT )S )T B S . 2 The Kronecker Product The Kronecker product is a binary matrix operator that maps two arbitrarily dimensioned matrices into a larger matrix with special block structure. Given the n × m matrix An×m and the p × q matrix Bp×q ⎡ ⎤ ⎡ ⎤ a1,1 . . . a1,m b1,1 . . . b1,q ⎢ ⎢ .. .. ... ⎥ ... ⎥ A = ⎣ ... B = ⎣ ... (5) ⎦ ⎦ . . an,1 ... an,m bp,1 n×m ... bp,q p×q their Kronecker product, denoted A ⊗ B, is the np × mq matrix with the block structure ⎡ ⎤ a1,1 B . . . a1,m B ⎢ ⎥ .. .. ... . A⊗B =⎣ ⎦ . . an,1 B . . . an,m B (6) np×mq For example, given � A= 1 2 0 −1 � � B2×3 = 2×2 1 2 3 4 5 6 � (7) 2×3 the Kronecker product A ⊗ B is ⎡ ⎤ 1 2 3 2 4 6 ⎢ 4 5 6 8 10 12 ⎥ ⎥ A⊗B =⎢ . ⎣ 0 0 0 −1 −2 −3 ⎦ 0 0 0 −4 −5 −6 4×6 2 (8) 2.1 Properties of the Kronecker Product Operator In the following it is assumed that A, B, C, and D are real valued matrices. Some identities only hold for appropriately dimensioned matrices. 1. The Kronecker product is a bilinear operator. Given α ∈ IR , A ⊗ (αB) = α(A ⊗ B) (αA) ⊗ B = α(A ⊗ B). (9) 2. Kronecker product distributes over addition: (A + B) ⊗ C = (A ⊗ C) + (B ⊗ C) A ⊗ (B + C) = (A ⊗ B) + (A ⊗ C). (10) 3. The Kronecker product is associative: (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C). (11) 4. The Kronecker product is not in general commutative, i.e. usually (A ⊗ B) �= (B ⊗ A). (12) 5. Transpose distributes over the Kronecker product (does not invert order) (A ⊗ B)T = AT ⊗ B T . (13) 6. Matrix multiplication, when dimensions are appropriate, (A ⊗ B)(C ⊗ D) = (AC ⊗ BD). (14) 7. When A and B are square and full rank (A ⊗ B)−1 = (A−1 ⊗ B −1 ). (15) 8. The determinant of a Kronecker product is (note right hand side exponents) det(An×n ⊗ Bm×m ) = det(A)m · det(B)n . (16) 9. The trace of a Kronecker product is trace(A ⊗ B) = trace(A) · trace(B). (17) 10. Stack of a matrix multiplication, when dimensions are appropriate for the product ABC to be well deﬁned, is (ABC)S = (C T ⊗ A)B S . (18) References [1] Alexander Graham. Kronecker Products and Matrix Calculus With Applications. Halsted Press,John Wiley and Sons, NY, 1981. 3

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