Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESClearly result (8.63) for diagonal matrices is a special case of this result. Moreover,it may be shown that the inverse of a non-singular lower (upper) triangular matrixis also lower (upper) triangular.8.12.3 Symmetric and antisymmetric matricesA square matrix A of order N with the property A = A T is said to be symmetric.Similarly a matrix for which A = −A T is said to be anti- or skew-symmetricand its diagonal elements a 11 ,a 22 ,...,a NN are necessarily zero. Moreover, if A is(anti-)symmetric then so too is its inverse A −1 . This is easily proved by notingthat if A = ±A T then(A −1 ) T =(A T ) −1 = ±A −1 .Any N × N matrix A can be written as the sum of a symmetric and anantisymmetric matrix, since we may writeA = 1 2 (A + AT )+ 1 2 (A − AT )=B + C,where clearly B = B T and C = −C T . The matrix B is therefore called thesymmetric part of A, andC is the antisymmetric part.◮If A is an N × N antisymmetric matrix, show that |A| =0if N is odd.If A is antisymmetric then A T = −A. Using the properties of determinants (8.49) and(8.51), we have|A| = |A T | = |−A| =(−1) N |A|.Thus, if N is odd then |A| = −|A|, which implies that |A| =0.◭8.12.4 Orthogonal matricesA non-singular matrix with the property that its transpose is also its inverse,A T = A −1 , (8.65)is called an orthogonal matrix. It follows immediately that the inverse of anorthogonal matrix is also orthogonal, since(A −1 ) T =(A T ) −1 =(A −1 ) −1 .Moreover, since for an orthogonal matrix A T A = I, we have|A T A| = |A T ||A| = |A| 2 = |I| =1.Thus the determinant of an orthogonal matrix must be |A| = ±1.An orthogonal matrix represents, in a particular basis, a linear operator thatleaves the norms (lengths) of real vectors unchanged, as we will now show.270