Free Singular Values Calculator

What Are Singular Values?

For any $m \times n$ matrix $A$, its singular values are defined as the square roots of the eigenvalues of the matrix $A^*A$, where $A^*$ denotes the conjugate transpose (or simply the transpose if $A$ contains only real entries). Because $A^*A$ is a positive semi‑definite matrix, all its eigenvalues are non‑negative, so the square roots are always well‑defined and real.

The singular values are usually denoted by $\sigma_1, \sigma_2, \dots, \sigma_r$ (where $r$ is the rank of $A$) and are listed in descending order. The largest singular value $\sigma_1$ gives the operator norm of $A$ with respect to the Euclidean norm.

How to Use the Singular Values Calculator

This tool makes finding the singular values of a matrix straightforward. You only need to:

1. Specify the number of rows and columns (the matrix size).
2. Enter the numerical entries of your matrix into the cells.
3. The calculator instantly displays the computed singular values at the bottom.

You can test the tool with various matrices – symmetric, diagonal, or rectangular – and observe how the singular values change.

Manual Computation of Singular Values

If you prefer to compute matrix singular values by hand, follow these steps:

1. Form $A^*$: Obtain the Hermitian (conjugate) transpose of $A$. For real matrices, this is simply the transpose.
2. Multiply $A^*A$: Perform matrix multiplication to get an $n \times n$ matrix.
3. Find eigenvalues: Solve the characteristic polynomial of $A^*A$ to get its eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$.
4. Take square roots: The singular values are $\sigma_i = \sqrt{\lambda_i}$ (only the non‑negative ones matter, as eigenvalues of $A^*A$ are non‑negative).

In practice, the standard approach uses the Singular Value Decomposition (SVD), which factorizes $A$ into $U \Sigma V^*$ where $\Sigma$ contains the singular values along its diagonal. Many linear algebra calculators incorporate the SVD to deliver results quickly.

Singular Values vs. Eigenvalues

Although both concepts are related, they are not the same:

• Applicability: Every matrix, regardless of shape, has singular values. Eigenvalues exist only for square matrices.
• Nature: Singular values are always real and non‑negative. Eigenvalues can be negative, zero, or even complex (for non‑symmetric matrices).

For a diagonal matrix, the singular values equal the absolute values of the diagonal entries. For a symmetric matrix that is also positive semi‑definite, the singular values coincide with the eigenvalues; otherwise, they equal the absolute values of the eigenvalues.

These properties make singular value decomposition a powerful tool for dimensionality reduction, noise filtering, and solving linear systems, beyond what eigenvalue decomposition can offer for non‑square matrices.