Mathematical Methods for Physics and Engineering - Matematica.NET

8.11 THE RANK OF A MATRIX8.11 The rank of a matrixThe rank of a general M × N matrix is an important concept, particularly inthe solution of sets of simultaneous linear equations, to be discussed in the nextsection, and we now discuss it in some detail. Like the trace and determinant,the rank of matrix A is a single number (or algebraic expression) that dependson the elements of A. Unlike the trace and determinant, however, the rank of amatrix can be defined even when A is not square. As we shall see, there are twoequivalent definitions of the rank of a general matrix.Firstly, the rank of a matrix may be defined in terms of the linear independenceof vectors. Suppose that the columns of an M × N matrix are interpreted asthe components in a given basis of N (M-component) vectors v 1 , v 2 ,...,v N ,asfollows:⎛A = ⎝↑ ↑ ↑v 1 v 2 ... v N↓ ↓ ↓Then the rank of A, denoted by rank A or by R(A), is defined as the numberof linearly independent vectors in the set v 1 , v 2 ,...,v N , and equals the dimensionof the vector space spanned by those vectors. Alternatively, we may consider therows of A to contain the components in a given basis of the M (N-component)vectors w 1 , w 2 ,...,w M as follows:⎛A = ⎜⎝← w 1 →← w 2 →.← w M →It may then be shown § that the rank of A is also equal to the number oflinearly independent vectors in the set w 1 , w 2 ,...,w M . From this definition it isshould be clear that the rank of A is unaffected by the exchange of two rows(or two columns) or by the multiplication of a row (or column) by a constant.Furthermore, suppose that a constant multiple of one row (column) is added toanother row (column): for example, we might replace the row w i by w i + cw j .This also has no effect on the number of linearly independent rows and so leavesthe rank of A unchanged. We may use these properties to evaluate the rank of agiven matrix.A second (equivalent) definition of the rank of a matrix may be given and usesthe concept of submatrices. A submatrix of A is any matrix that can be formedfrom the elements of A by ignoring one, or more than one, row or column. It⎞⎠ .⎞⎟⎠ .§ For a fuller discussion, see, for example, C. D. Cantrell, Modern Mathematical Methods for Physicistsand Engineers (Cambridge: Cambridge University Press, 2000), chapter 6.267