Matrix Norms

Matrix Norms

A matrix norm $\|A\|$ measures the "size" of a matrix. Different norms capture different aspects of the matrix.

Frobenius Norm

The most common: sum of squares of all entries, then square root.

$\|A\|_F = \sqrt{\sum_{i,j} a_{ij}^2}$

Analogous to the Euclidean length of the vector of all entries.

Infinity Norm (Max Row Sum)

The maximum absolute row sum — the worst-case amplification when multiplying a vector with $\|\mathbf{v}\|_\infty \leq 1$.

$\|A\|_\infty = \max_i \sum_j |a_{ij}|$

1-Norm (Max Column Sum)

The maximum absolute column sum.

$\|A\|_1 = \max_j \sum_i |a_{ij}|$

Example

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

$\|A\|_F = \sqrt{1+4+9+16} = \sqrt{30} \approx 5.4772$

$\|A\|_\infty = \max(1+2,\ 3+4) = 7.0000, \qquad \|A\|_1 = \max(1+3,\ 2+4) = 6.0000$

Your Task

Implement frobenius_norm(A), infinity_norm(A), and one_norm(A).