Singular Value Decomposition Calculator

Math Linear Algebra • Linear Transformations and Eigenvalues

Written by STEM Calculators Team

Compute the Singular Value Decomposition (SVD) \(A=U\Sigma V^T\) for rectangular matrices. Singular values \(\sigma_i\) come from eigenvalues of \(A^TA\), rank is the count of nonzero \(\sigma_i\), and truncated SVD gives the best rank-\(k\) approximation.

Rows \(m\)

Cols \(n\)

Tolerance

Display decimals

SVD mode

Steps

Show singular value plot Show truncated rank-\(k\) approximation Show orthonormality/reconstruction checks

Matrix \(A\)

A is 3×2

Inputs accept -3.5, 2e-4, fractions like 7/3, and constants pi, e.

Results

Singular values \(\sigma_1\ge\sigma_2\ge\cdots\)

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Estimated rank

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Shape summary

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Threshold for “nonzero” \(\sigma\)

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Decomposition

Displayed matrices depend on mode (reduced/full). \(V^T\) is shown.

Checks

Orthonormality and reconstruction error.

Ready

Enter matrix \(A\), then click “Calculate”.

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Theory

Singular Value Decomposition (SVD)

The Singular Value Decomposition factors any real matrix \(A\in\mathbb{R}^{m\times n}\) as \[ A = U\Sigma V^T, \] where \(U\in\mathbb{R}^{m\times m}\) and \(V\in\mathbb{R}^{n\times n}\) are orthogonal matrices (\(U^TU=I\), \(V^TV=I\)), and \(\Sigma\in\mathbb{R}^{m\times n}\) is a diagonal matrix (all off-diagonal entries are \(0\)) whose diagonal entries are nonnegative and sorted: \[ \sigma_1 \ge \sigma_2 \ge \cdots \ge 0. \] The numbers \(\sigma_i\) are the singular values. They measure how strongly the linear transformation represented by \(A\) stretches vectors in special directions.

A helpful geometric picture is: \(V^T\) rotates or reflects the input space, \(\Sigma\) scales along coordinate axes, and \(U\) rotates or reflects the result. In other words, \(A\) maps the unit sphere to an ellipsoid, and the ellipsoid’s semi-axis lengths are the singular values \(\sigma_i\).

Reduced (economy) vs full SVD

Let \(p=\min(m,n)\). The reduced (economy) SVD is \[ A = U_p \Sigma_p V_p^T, \] where \(U_p\in\mathbb{R}^{m\times p}\), \(\Sigma_p\in\mathbb{R}^{p\times p}\), and \(V_p\in\mathbb{R}^{n\times p}\). This form is often preferred because it avoids extra “padding” columns when \(A\) is rectangular.

The full SVD uses \(U\in\mathbb{R}^{m\times m}\), \(V\in\mathbb{R}^{n\times n}\), and \(\Sigma\in\mathbb{R}^{m\times n}\). The additional columns of \(U\) and \(V\) complete orthonormal bases in \(\mathbb{R}^m\) and \(\mathbb{R}^n\).

How the SVD is computed from \(A^TA\)

A standard route to the SVD is through the symmetric matrix \(A^TA\in\mathbb{R}^{n\times n}\). It is always symmetric and positive semidefinite, so it has real eigenvalues \(\mu_i\ge 0\) and an orthonormal eigenbasis: \[ A^TAv_i = \mu_i v_i,\qquad v_i^Tv_j=\delta_{ij}. \] The right singular vectors are the eigenvectors \(v_i\), and the singular values are \[ \sigma_i = \sqrt{\mu_i}. \] When \(\sigma_i>0\), the corresponding left singular vector is obtained by \[ u_i = \frac{Av_i}{\sigma_i}. \] This guarantees \(u_i\) has unit length (up to rounding) and that the decomposition \(A=\sum_i \sigma_i u_i v_i^T\) holds.

Numerically, very small eigenvalues can appear as tiny negative numbers due to rounding; these are treated as \(0\) before taking square roots. A tolerance is also used to decide which \(\sigma_i\) count as “nonzero.”

Rank, nullity, and conditioning

The number of nonzero singular values equals the rank: \[ \mathrm{rank}(A)=\#\{i:\sigma_i>0\}. \] For an \(m\times n\) matrix, the nullity is \(n-\mathrm{rank}(A)\). Singular values also describe numerical stability. If \(\sigma_{\min}\) is very small compared to \(\sigma_{\max}\), the matrix is close to rank-deficient and computations such as solving least-squares problems may be sensitive to noise. A common indicator is the condition number \[ \kappa(A)=\frac{\sigma_{\max}}{\sigma_{\min}} \] (defined when \(\sigma_{\min}>0\)).

Truncated SVD and best low-rank approximation

Keeping only the largest \(k\) singular values gives the truncated SVD: \[ A_k=\sum_{i=1}^k \sigma_i u_i v_i^T. \] A key theorem states that \(A_k\) is the best rank-\(k\) approximation to \(A\) in Frobenius norm: \[ \|A-A_k\|_F = \min_{\mathrm{rank}(B)\le k}\|A-B\|_F. \] This is why SVD appears in image compression, denoising, and dimensionality reduction: the largest singular values capture the dominant structure, while smaller ones often correspond to noise or less important variation.

Closely related ideas appear in PCA: if the columns of \(A\) are centered data, the right singular vectors identify principal directions.

University note: pseudoinverse and least squares

The Moore–Penrose pseudoinverse is built from the SVD: \[ A^+ = V\Sigma^+U^T, \] where \(\Sigma^+\) replaces each nonzero \(\sigma_i\) by \(1/\sigma_i\). This produces stable least-squares solutions and explains how “small” singular values amplify noise (because \(1/\sigma_i\) becomes large).

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