8 1. Preliminaries 1.7. Transposes For an m × n matrix A, the transpose AT is the n × m matrix which has aij in position (j, i). Thus, row i of A becomes column i of AT , and column j of A becomes row j of AT . You can think of AT as the array obtained by flipping A across its main diagonal. Note that (AT )T = A. We say a square matrix A is symmetric if AT = A, and skew-symmetric if AT = −A. Since (A + B)T = AT + BT and (cA)T = cAT , for any scalar c, we see that A + AT is symmetric and A AT is skew-symmetric. If char(F) = 2, we have (1.5) A = A + AT 2 + A AT 2 , which expresses a general square matrix A as the sum of a symmetric matrix and a skew-symmetric matrix. If A is a complex matrix, then A∗ denotes the conjugate transpose i.e., the (j, i) entry of A∗ is aij. A square matrix A is Hermitian if A∗ = A, and it is skew-Hermitian if A∗ = −A. Note that A is Hermitian if and only if iA is skew- Hermitian. Analogous to equation (1.5), we have A = A+A∗ 2 + A−A∗ 2 . If we set H = A+A∗ 2 and K = A−A∗ 2i , then H and K are both Hermitian and (1.6) A = A + A∗ 2 + i A A∗ 2i = H + iK. Equation (1.6) is analagous to writing a complex number z as z = x + iy, where x and y are real numbers. Direct computation shows that (AB)T = BT AT and (AB)∗ = B∗A∗. For an invertible matrix A we have (AT )−1 = (A−1)T and (A∗)−1 = (A−1)∗. 1.8. Special Types of Matrices We say a square matrix D is diagonal if dij = 0 for all i = j i.e., if all off-diagonal entries are zero. In this case we often denote the ith diagonal entry as di thus D = d1 0 · · · 0 0 d2 · · · 0 . . . . ... . . 0 0 · · · dn . We shall also abbreviate this as D = diag(d1,...,dn). If A is an m × n matrix, then column j of the product AD is dj(Aj), where Aj denotes the jth column of A. Thus, the effect of multiplying a matrix A on the right by a diagonal matrix D is to multiply the columns of A by the corresponding diagonal entries of D. If A is n × k, then the ith row of DA is di times the ith row of A—multiplying a matrix A on the left by a diagonal matrix D multiplies the rows of A by the corresponding diagonal entries of D.
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