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8) Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 26 The next statement seems to be important for understanding the internal relationship within different classes of matrices. 1. 1. Hermitian and real orthogonal, symmetric and skew-symmetric matrix is normal, that is, it satisfies the condition AA* = A*A for complex matrices and A A r = A r A for real matrices 2. A matrix A is normal if and only if the matrices A and A* have the same eigenvectors. 3. 1. If Z [a"a2,a22a'2a23a'3] Eb,, ,21 then allbll a21bll A| al2bl2 a13bll azzb12 az3bll alibi2 a12bll azlb12 a22bll a13b12] az3b12 J The following properties for the Kronecker product are fulfilled: (a) A I,• | = O 9 9 o o O IA i 9 9 9 0 9 A O = diag [A, A .
7. A matrix A 9 R nxn is said to be o r t h o g o n a l if A r A = A A v = I~x~, or, equivalently, if A v = A -1 and unitary if A 9 C ~xn and A * A = A A * = l~x~, or, equivalently, if A * = A -1. 2 Some matrix properties The following matrix properties hold: 1. Commutativity of the summing operation, that is, A+B--B+A 2. Associativity of the summing operation, that is, (A + B) + C = A + (B + C) 3. Associativity of the multiplication operation, that is, (AB) C = A (BC) 4. Distributivity of the multiplication operation with respect to the summation operation, that is, (A + B) C = A C + B C , C (A + B) = C A + C B AI=IA--A Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 22 m,n 5.
P). 28 Advanced Mathematical Toolsfor Automatic Control Engineers: Volume 1 (A | B) -1 -- A -1 @ B -1 ] provided that both A -1 and B -1 exist. 2. x. 2. ) - det [pT (I~x. x. x~ | A) = det (diag (A, A . . . 4 Submatrices, partitioning of matrices and Schur's formulas Given matrix A _ [aij]i,j=l , m , n if a number of complete rows or columns of A are deleted, or if some complete rows or complete columns are deleted, the new matrix that is obtained is called a submatrix of A. A division of matrices into submatrices is referred to as a partition of the matrix.