SVD Calculator

Singular Value Decomposition (SVD)

Any real matrix $A$ of size $m \times n$ can be decomposed as:

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

Here $U$ is an $m \times m$ orthogonal matrix, $\Sigma$ is an $m \times n$ diagonal matrix of non-negative singular values, and $V^T$ is the transpose of an $n \times n$ orthogonal matrix.

Applications

SVD is used in data compression, principal component analysis, solving least-squares problems, computing matrix pseudoinverses, and determining matrix rank and condition number.

Frequently Asked Questions

What are singular values?

Singular values are the non-negative square roots of the eigenvalues of $A^T A$ (or $A A^T$). They appear on the diagonal of $\Sigma$.

Is SVD unique?

The singular values are unique (when sorted in descending order), but the matrices $U$ and $V$ are generally not unique.

How is SVD related to eigenvalues?

The columns of $V$ are eigenvectors of $A^T A$, and the singular values are the square roots of the corresponding eigenvalues.

Can SVD handle rectangular matrices?

Yes. SVD works for any $m \times n$ matrix, whether square or rectangular, full rank or rank-deficient.

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Singular Value Decomposition (SVD)

Applications

Frequently Asked Questions

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