Rank of Matrix

Definition

$\mathrm{rank}(\mathbf{A})$

The rank of matrix is the number of linearly independent column vectors, or the number of non-zero pivots in Gauss Elimination

Facts

$\mathrm{rank}(AB)\le \min (\mathrm{rank}(A),\mathrm{rank}(B))$

For the non-singular matrices $P$ and $Q$, $\mathrm{rank}(PAQ)=\mathrm{rank}(A)$

Rank–Nullity Theorem

$\mathrm{rank}(A)=\mathrm{rank}(A^{⊺})=\mathrm{rank}(A^{⊺}A)=\mathrm{rank}(A{A}^{⊺})$

If a matrix $A$ is Symmetric Matrix, then $\mathrm{rank}(A)$ is the number of non-zero eigenvalues.