Lecture 9: Equivalent Matrices

9.1 Equivalent Matrices

Definition: equivalent-matrices

Let $A$ and $B$ both be $m\times n$ matrices. If an invertible $m\times m$ matrix $P$ and an invertible $n\times n$ matrix $Q$ exist such that

then $A$ and $B$ are equivalent, written $A\sim B$.

Left multiplication by an invertible matrix represents a finite sequence of elementary row operations, while right multiplication represents a finite sequence of elementary column operations. Thus two matrices are equivalent precisely when one can be transformed into the other by finitely many elementary row and column operations.

9.2 Rank Normal Form

Definition: rank-normal-form

If $\operatorname{rank}(A)=r$, invertible matrices $P$ and $Q$ exist such that

The matrix on the right is the rank normal form of $A$, where $I_r$ is the $r\times r$ identity matrix. The sizes of the zero blocks are determined by the numbers $m$ and $n$ of rows and columns of $A$.

Conclusions

• Every $m\times n$ matrix is equivalent to a rank normal form of this type.
• The rank normal form is uniquely determined by the matrix dimensions and its rank.
• Two matrices of the same size are equivalent if and only if they have the same rank:

• When reducing a matrix to rank normal form, $P$ records the row operations and $Q$ records the column operations.