SVD Calculator
Calculate the singular value decomposition of a real rectangular matrix. Find U, the diagonal singular-value matrix Σ, and V in the factorization A = UΣVᵀ.
Enter your matrix
Enter one row per line. Separate numbers with spaces, commas, or semicolons. Supports matrices up to 8 × 8.
What is singular value decomposition?
Singular value decomposition (SVD) expresses a real matrix as a product of matrices containing left singular vectors, singular values, and right singular vectors. It works for square and rectangular matrices.
- U: left singular vectors.
- Σ: nonnegative singular values in descending order.
- V: right singular vectors; its transpose is used in the product.
Applications of SVD
SVD is used in low-rank approximation, data compression, principal component analysis, least-squares problems, and numerical analysis. Singular values can also help estimate matrix rank and identify poorly conditioned problems.
Frequently asked questions
Can I enter a rectangular matrix?
Yes. This calculator supports rectangular matrices up to 8 rows and 8 columns.
Are singular values always positive?
Singular values are always nonnegative. A singular value can be zero when the matrix is rank deficient.
Why is the reconstruction error not exactly zero?
Computers use finite-precision arithmetic, so small rounding errors are normal.
What does a very small singular value mean?
It may indicate a direction with little variation or near-linear dependence among the matrix columns or rows.