Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESmay be shown that the rank of a general M × N matrix is equal to the size ofthe largest square submatrix of A whose determinant is non-zero. Therefore, if amatrix A has an r × r submatrix S with |S| ≠ 0, but no (r +1)× (r + 1) submatrixwith non-zero determinant then the rank of the matrix is r. From either definitionit is clear that the rank of A is less than or equal to the smaller of M and N.◮Determine the rank of the matrix⎛A = ⎝ 1 1 0 −22 0 2 24 1 3 1⎞⎠ .The largest possible square submatrices of A must be of dimension 3 × 3. Clearly, Apossesses four such submatrices, the determinants of which are given by∣ 1 1 0∣∣∣∣∣ 1 1 −22 0 2∣ 4 1 3 ∣ =0, 2 0 24 1 1 ∣ =0,∣1 0 −22 2 24 3 1∣ ∣∣∣∣∣ 1 0 −2∣ =0, 0 2 21 3 1∣ =0.(In each case the determinant may be evaluated as described in subsection 8.9.1.)The next largest square submatrices of A are of dimension 2 × 2. Consider, for example,the 2 × 2 submatrix formed by ignoring the third row and the third and fourth columnsof A; this has determinant∣ 1 12 0 ∣ =1× 0 − 2 × 1=−2.Since its determinant is non-zero, A is of rank 2 and we need not consider any other 2 × 2submatrix. ◭In the special case in which the matrix A is a square N×N matrix, by comparingeither of the above definitions of rank with our discussion of determinants insection 8.9, we see that |A| = 0 unless the rank of A is N. Inotherwords,A issingular unless R(A) =N.8.12 Special types of square matrixMatrices that are square, i.e. N × N, are very common in physical applications.We now consider some special forms of square matrix that are of particularimportance.8.12.1 Diagonal matricesThe unit matrix, which we have already encountered, is an example of a diagonalmatrix. Such matrices are characterised by having non-zero elements only on the268