Mathematical Methods for Physics and Engineering - Matematica.NET

8.12 SPECIAL TYPES OF SQUARE MATRIXSuppose that y = A x is represented in some coordinate system by the matrixequation y = Ax; then〈y|y〉 is given in this coordinate system byy T y = x T A T Ax = x T x.Hence 〈y|y〉 = 〈x|x〉, showing that the action of a linear operator represented byan orthogonal matrix does not change the norm of a real vector.8.12.5 Hermitian and anti-Hermitian matricesAn Hermitian matrix is one that satisfies A = A † ,whereA † is the Hermitian conjugatediscussed in section 8.7. Similarly if A † = −A, thenA is called anti-Hermitian.A real (anti-)symmetric matrix is a special case of an (anti-)Hermitian matrix, inwhich all the elements of the matrix are real. Also, if A is an (anti-)Hermitianmatrix then so too is its inverse A −1 ,since(A −1 ) † =(A † ) −1 = ±A −1 .Any N × N matrix A can be written as the sum of an Hermitian matrix andan anti-Hermitian matrix, sinceA = 1 2 (A + A† )+ 1 2 (A − A† )=B + C,where clearly B = B † and C = −C † . The matrix B is called the Hermitian part ofA, andC is called the anti-Hermitian part.8.12.6 Unitary matricesA unitary matrix A is defined as one for whichA † = A −1 . (8.66)Clearly, if A is real then A † = A T , showing that a real orthogonal matrix is aspecial case of a unitary matrix, one in which all the elements are real. We notethat the inverse A −1 of a unitary is also unitary, since(A −1 ) † =(A † ) −1 =(A −1 ) −1 .Moreover, since for a unitary matrix A † A = I, we have|A † A| = |A † ||A| = |A| ∗ |A| = |I| =1.Thus the determinant of a unitary matrix has unit modulus.A unitary matrix represents, in a particular basis, a linear operator that leavesthe norms (lengths) of complex vectors unchanged. If y = A x is represented insome coordinate system by the matrix equation y = Ax then 〈y|y〉 is given in thiscoordinate system byy † y = x † A † Ax = x † x.271