Linear Algebra

Section V. Change of Basis 239We can get some insight into the classes by comparing matrix equivalence withrow equivalence (recall that matrices are row equivalent when they can be reducedto each other by row operations). In Ĥ = P HQ, the matrices P andQ are nonsingular and thus each can be written as a product of elementaryreduction matrices (Lemma 4.8). Left-multiplication by the reduction matricesmaking up P has the effect of performing row operations. Right-multiplicationby the reduction matrices making up Q performs column operations. Therefore,matrix equivalence is a generalization of row equivalence — two matrices are rowequivalent if one can be converted to the other by a sequence of row reductionsteps, while two matrices are matrix equivalent if one can be converted to theother by a sequence of row reduction steps followed by a sequence of columnreduction steps.Thus, if matrices are row equivalent then they are also matrix equivalent(since we can take Q to be the identity matrix and so perform no columnoperations). The converse, however, does not hold.2.5 Example These two( 1) 0( 1) 10 0 0 0are matrix equivalent because the second can be reduced to the first by thecolumn operation of taking −1 times the first column and adding to the second.They are not row equivalent because they have different reduced echelon forms(in fact, both are already in reduced form).We will close this section by finding a set of representatives for the matrixequivalence classes. ∗2.6 Theorem Any m×n matrix of rank k is matrix equivalent to the m×nmatrix that is all zeros except that the first k diagonal entries are ones.⎛⎞1 0 . . . 0 0 . . . 00 1 . . . 0 0 . . . 0.0 0 . . . 1 0 . . . 0⎜0 0 . . . 0 0 . . . 0⎟⎜⎝.0 0 . . . 0 0 . . . 0Sometimes this is described as a block partial-identity form.( ) I Z⎟⎠ZZ∗ More information on class representatives is in the appendix.