Singular Values Calculator

What Are Singular Values?

In linear algebra, the singular values of an $m \times n$ real or complex matrix $A$ are the non-negative square roots of the eigenvalues of the associated symmetric positive semi-definite matrix $A^T A$ (or $A A^T$). They represent the lengths of the principal semi-axes of the hyperellipsoid that results from mapping the unit sphere through the linear transformation $A$.

Singular values are central to the Singular Value Decomposition (SVD), one of the most powerful and ubiquitous matrix factorizations in modern applied mathematics, data science, machine learning, and quantum mechanics. Explore related matrix analysis with our Pseudoinverse Calculator, Matrix Determinant Calculator, and Matrix Norm Calculator.

Singular Value Decomposition (SVD) Formula

Any real $m \times n$ matrix $A$ can be factored as:

$$A = U \Sigma V^T$$

Where:

• U is an $m \times m$ orthogonal matrix whose columns are the left-singular vectors of $A$ (eigenvectors of $A A^T$).
• Σ is an $m \times n$ rectangular diagonal matrix containing the singular values $\sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_k \ge 0$ on its main diagonal.
• V is an $n \times n$ orthogonal matrix whose columns are the right-singular vectors of $A$ (eigenvectors of $A^T A$).

How to Calculate Singular Values Step-by-Step

1. Form the symmetric product: Compute $S = A^T A$ if $m \ge n$ (an $n \times n$ matrix) or $S = A A^T$ if $m < n$ (an $m \times m$ matrix).
2. Find the eigenvalues of S: Solve the characteristic polynomial $\det(S - \lambda I) = 0$ to find the roots $\lambda_1 \ge \lambda_2 \ge \dots \ge 0$. Because $S$ is symmetric positive semi-definite, all its eigenvalues are guaranteed to be real and non-negative.
3. Take the square root: Compute $\sigma_i = \sqrt{\lambda_i}$. The resulting values ordered from greatest to least are the singular values of $A$.

Key Matrix Properties Derived from Singular Values

• Matrix Rank: The numerical rank of matrix $A$ is equal to the number of non-zero singular values.
• Spectral 2-Norm: The induced Euclidean matrix norm $\|A\|_2$ is equal to the largest singular value: $\|A\|_2 = \sigma_1 = \sigma_{\max}$.
• Frobenius Norm: The root-sum-of-squares of all matrix entries equals the Euclidean length of the singular value vector: $\|A\|_F = \sqrt{\sigma_1^2 + \sigma_2^2 + \dots + \sigma_k^2}$.
• Nuclear (Trace) Norm: The sum of all singular values $\|A\|_* = \sum \sigma_i$ plays a fundamental role in low-rank matrix completion and compressed sensing.
• Condition Number: The ratio $\kappa(A) = \sigma_{\max} / \sigma_{\min}$ measures how sensitive linear systems $Ax = b$ are to numerical errors or input perturbations. A matrix with an infinite condition number is singular (non-invertible).