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Theorem
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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.
