Full rank matrices

Alicia Y. Tsai; Authors: Laurent EI Ghaoui; Contributors: Phan Hoang, Tony Tin, Ha To Tam, Nguyen Van Kep, Le Dinh Nam, Juliette Decugis, Nguyen Quoc Cuong.; Giuseppe C. Calafiore

Full rank matrices

Theorem

A matrix $A$ in $\mathbb{R}^{m \times n}$ is:

● Full column rank if and only if $A^T A$ is invertible.

● Full row rank if and only if $AA^T$ is invertible.

Proof:

The matrix is full column rank if and only if its nullspace is reduced to the singleton $\{0\}$ , that is,

$Ax = 0 \implies x = 0$

If $A^T A$ is invertible, then the condition $Ax = 0$ implies $A^T A x = 0$ , which in turn implies $x = 0$ .

Conversely, assume that the matrix is full column rank, and let $x$ be such that $A^T A x = 0$ . We then have $x^T A^T A x = ||Ax||_2^2 = 0$ , which means $Ax = 0$ . Since $A$ is full column rank, we obtain $x = 0$ , as desired. The proof for the other property follows similar lines.

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Theorem

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