Matrix norm

Arguably, the most common norm for a matrix is the spectral norm, which is defined for a matrix $$A \in \mathbb{R}^{n\times m}$$ as

$$||A||:= \max_{x\neq 0}\frac{||Ax||}{||x||}$$.

Using the upper bound on a quadratic form of a symmetric matrix (see Quadratic form), it is easy to show that $$||A|| = \sqrt{\lambda_{\max}(A^TA)}$$, where $$\lambda_{\max}(A^T A)$$ is the largest eigenvalue of the (symmetric) matrix $$A^T A$$:

$$||A||{}^2 = \max_{x\neq 0}\frac{||Ax||{}^2}{||x||{}^2} = \max_{x\neq 0}\frac{x^T A^T A x}{||x||{}^2} = \lambda_{\max}(A^T A) $$.

Intuitive explanation
The matrix norm is the matrix norm tells us how "large" a matrix is. This is understood intuitively when matrix norm is thought as the (vector) norm $$||Au||$$ for the unit vector $$u$$ that maximizes the gain of a matrix.