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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2015 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.