Frobenius norm via singular values
The Frobenius norm is the most down-to-earth way to measure a matrix: pretend its entries are one long vector and take the ordinary Euclidean length. ||A||_F = sqrt(sum of all a_ij^2). It is the total energy of the matrix, treating every entry on equal footing — a single number for how big the whole array is.
Through the SVD it gets a cleaner face: ||A||_F = sqrt(sigma_1^2 + sigma_2^2 + ... + sigma_r^2). The sum of squared entries equals the sum of squared singular values. This is the Schatten 2-norm, the partner of the spectral norm (Schatten infinity) and the nuclear norm (Schatten 1) — the same root-sum-of-squares applied to the singular values instead of the raw entries.
The deep reason this works is rotation invariance. For any orthogonal U and V, ||U A V^T||_F = ||A||_F — multiplying by orthogonal matrices does not change the Frobenius norm, because orthogonal maps preserve lengths. Since A = U Sigma V^T, the norm of A equals the norm of Sigma, and Sigma's only nonzero entries are the singular values down the diagonal. Hence the squared-singular-value formula.
Equivalently, ||A||_F^2 = trace(A^T A) = sum of eigenvalues of A^T A = sum of sigma_i^2 — three ways to say the same energy. This is the norm that the Frobenius case of Eckart-Young minimizes, and the natural measure of approximation error when you care about overall fit rather than worst-case stretch.
Three equal forms: entrywise energy, singular-value energy, and the trace of A^T A.
Rotation invariance is the headline: ||A||_F depends only on the singular values, not on the orientation of A's rows or columns. So it is blind to any orthogonal change of input or output basis.