Linear and multilinear algebra; matrix theory Canonical forms and decompositions Theorem

Eckart–Young theorem

Builds on Singular value decomposition Singular value Rank of a matrix Frobenius norm Operator norm

Statement

Let $A$ be an $m \times n$ complex matrix with singular value decomposition $A = \sum_{i=1}^{r} \sigma_i u_i v_i^*$ . Here $r$ is the rank of $A$ , $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_r > 0$ are its singular values, and $u_1, \dots, u_r$ and $v_1, \dots, v_r$ are orthonormal families in $\C^m$ and $\C^n$ . Let $0 \le k < r$ and put $A_k = \sum_{i=1}^{k} \sigma_i u_i v_i^*$ . Then for every $m \times n$ complex matrix $B$ of rank at most $k$ :

$\norm{A - B}_2 \ge \norm{A - A_k}_2 = \sigma_{k+1}$ , where $\norm{\cdot}_2$ is the operator norm for the Euclidean norms.
$\norm{A - B}_F \ge \norm{A - A_k}_F = \left(\sigma_{k+1}^2 + \cdots + \sigma_r^2\right)^{1/2}$ , where $\norm{\cdot}_F$ is the Frobenius norm.

Remarks

For the operator norm, the kernel of $B$ has dimension at least $n - k$ , so it meets the span of $v_1, \dots, v_{k+1}$ in a unit vector $x$ . Then $\norm{(A - B)x} = \norm{Ax} \ge \sigma_{k+1}$ . The Frobenius case follows from Weyl's inequalities for singular values, $\sigma_{i+j-1}(A) \le \sigma_i(A - B) + \sigma_j(B)$ , applied with $j = k+1$ . For example, if $A = \operatorname{diag}(3, 2, 1)$ , the best rank one approximation is $\operatorname{diag}(3, 0, 0)$ , with operator norm error $2$ and Frobenius error $\sqrt{5}$ . The minimiser $A_k$ is unique in the Frobenius norm when $\sigma_k > \sigma_{k+1}$ ; in the operator norm it is not unique in general. The same result holds for real matrices with real singular value decompositions.

Going deeper

For a student

This is the theoretical basis of principal component analysis and of truncated SVD compression. Mirsky extended it to every unitarily invariant norm, so it is often called the Eckart–Young–Mirsky theorem. It also shows that $\sigma_{k+1}$ is the distance from $A$ to the set of matrices of rank at most $k$ . In particular, for a square invertible $A$ , the smallest singular value is the distance to the singular matrices.

Reading path

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Frobenius norm concept
Identity matrix object
Matrix definition
Operator norm concept
Rank of a matrix concept
Singular value concept
Vector space definition
Square matrix definition
Transpose definition
Linear map definition
Conjugate transpose definition
Endomorphism definition
Hermitian matrix definition
Unitary matrix definition
Eigenvalue definition
Positive semidefinite matrix definition
Gram matrices are positive semidefinite statement
Singular value definition
Singular value decomposition statement
Eckart–Young theorem statement

Tag	`JJT6GC055J`
URI	`m2://JJT6GC055J`
Kind	statement, theorem
Writing	W0 stub
MSC	15A18, 15A60, 65F55
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About	Singular value decomposition, Rank of a matrix
Uses	Singular value decomposition, Singular value, Rank of a matrix, Frobenius norm, Operator norm
Changed	2 November 2026