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Extra resources for Advanced Mathematical Tools for Control Engineers: Deterministic Systems
Some matrix properties . . . . . . . . . . Kronecker product . . . . . . . . . . . Submatrices, partitioning of matrices and Schur’s formulas Elementary transformations on matrices . . . . . Rank of a matrix . . . . . . . . . . . Trace of a quadratic matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1). Here the basic properties of matrices and the operations with them will be considered.
15). 14. For any A ∈ Rm×n and B ∈ Rn×p (AB) = B A Indeed, m,p n (AB) = = aj k bki k=1 i,j =1 m,p n 15. For any A ∈ Rn×n adj A = (adj A) adjIn×n = In×n adj (αA) = α n−1 adj A for any α ∈ F 16. For any A ∈ Cn×n adj A∗ = (adj A)∗ =B A bki aj k k=1 i,j =1 Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 24 17. 10) that leads to a matrix with the number det A in each position on its main diagonal and zeros elsewhere. 18. 4). 4). 19. If det A = 0, then A−1 = (A )−1 Indeed, In×n = AA−1 = AA−1 = A−1 A So, by definitions, A−1 = (A )−1 .
A certain number of inversions associated with a given permutation (j1 , j2 , . . , jn ) denoted briefly by t (j1 , j2 , . . , jn ). 3 Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 4 Clearly, there exists exactly n! = 1 · 2 · · · n permutations. 1. (1, 3, 2), (3, 1, 2), (3, 2, 1), (1, 2, 3), (2, 1, 3), (2, 3, 1) are the permutations of 1, 2, 3. 2. t (2, 4, 3, 1, 5) = 4. 3. A diagonal of an arbitrary square matrix A ∈ Rn×n is a sequence of elements of this matrix containing one and only one element from each row and one and only one element from each column.