Equivalent Matrix

An $m\times n$ Matrix ${\hbox{\sf A}}$ is said to be equivalent to another $m\times n$ Matrix ${\hbox{\sf B}}$ Iff

$${\hbox{\sf B}}={\hbox{\sf P}}{\hbox{\sf A}}{\hbox{\sf Q}}$$

for ${\hbox{\sf P}}$ and ${\hbox{\sf Q}}$ any $m\times n$ and $n\times n$ Matrices, respectively.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1103, 1979.