Basic Matrix Theory by Leonard E. Fuller

Author:Leonard E. Fuller
Language: eng
Format: azw3
Publisher: Dover Publications
Published: 2017-02-15T00:00:00+00:00

4.6 Rank of a Matrix

In the previous chapter, the rank of a matrix was defined in terms of the number of nonzero vectors of its canonical form. There is an older definition of rank in terms of determinants.

Definition 4.5. The rank of a matrix A is the order of its largest nonzero minor.

In this definition, all minors of this size may be nonzero, only some may be nonzero, or only the one may be nonzero.

In order to use this definition, one must have a way to find the largest nonzero minor. There are two ways to do this. One is to start with the largest minor which would be | A |, and then consider successively smaller minors. The other way is to start with the smallest minors and then consider successively larger minors. In the procedure, one notes that the rank is at least one if there is any nonzero element in the matrix, that is, if it is a nonzero matrix. Next, the 2×2 submatrices could be checked at sight to see if there is one with nonzero determinant; if so, the rank is at least two. The next step requires checking to see if any 3×3 submatrix has a nonzero determinant. The definition requires all of these determinants to be zero when the rank is two, but only one has to be nonzero if the rank is at least three. However, for the simple 4×4 there are sixteen 3×3 submatrices to be checked when the rank is two. Fortunately, not all of these will have to be evaluated. It can be proved that if some t × t submatrix has a nonzero determinant and all (t + 1) X (t + 1) submatrices that contain this t × t submatrix have a zero determinant, all ( t + 1) × (t + 1) submatrices have zero determinant. This means that the matrix must be of rank t by the definition.

This can be illustrated better using a numerical example. Consider the following matrix A whose rank is to be found.

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