Mathematical Methods for Physics and Engineering - Matematica.NET

8.12 SPECIAL TYPES OF SQUARE MATRIXleading diagonal, i.e. only elements A ij with i = j may be non-zero. For example,⎛1 0 0⎞A = ⎝ 0 2 0 ⎠ ,0 0 −3is a 3 × 3 diagonal matrix. Such a matrix is often denoted by A = diag (1, 2, −3).By performing a Laplace expansion, it is easily shown that the determinant of anN × N diagonal matrix is equal to the product of the diagonal elements. Thus, ifthe matrix has the form A = diag(A 11 ,A 22 ,...,A NN )then|A| = A 11 A 22 ···A NN . (8.63)Moreover, it is also straightforward to show that the inverse of A is also adiagonal matrix given by( )1A −1 1 1= diag , ,..., .A 11 A 22 A NNFinally, we note that, if two matrices A and B are both diagonal then they havethe useful property that their product is commutative:This is not true for matrices in general.AB = BA.8.12.2 Lower and upper triangular matricesA square matrix A is called lower triangular if all the elements above the principaldiagonal are zero. For example, the general form for a 3 × 3 lower triangularmatrix is⎛A 11 0 0⎞A 31 A 32 A 33A = ⎝ A 21 A 22 0 ⎠ ,where the elements A ij may be zero or non-zero. Similarly an upper triangularsquare matrix is one for which all the elements below the principal diagonal arezero. The general 3 × 3 form is thus⎛A 11 A 12 A 13⎞0 0 A 33A = ⎝ 0 A 22 A 23⎠ .By performing a Laplace expansion, it is straightforward to show that, in thegeneral N × N case, the determinant of an upper or lower triangular matrix isequal to the product of its diagonal elements,|A| = A 11 A 22 ···A NN . (8.64)269