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Key Features
Instant Calculation
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Singular Value Decomposition (SVD) factors any matrix A — square or rectangular — into A = UΣVᵀ, where U and V are orthogonal matrices and Σ holds the singular values (always non-negative, in descending order). Unlike eigendecomposition, SVD works for every matrix, which is why it's the workhorse behind PCA, image compression, and recommendation systems.
This calculator computes the full SVD for any matrix up to 6×6, derived from the eigendecomposition of AᵀA.
How it works
Enter a matrix (rows and columns can differ). The tool finds the eigenvalues and eigenvectors of AᵀA using the Jacobi method — the square roots of those eigenvalues are the singular values, the eigenvectors become V, and U is recovered by applying A to each column of V and rescaling.
- Click Calculate to see your results.
Examples
A 2×3 example
For A = [[3,2,2],[2,3,−2]], the singular values are 5, 3, and 0 — the zero singular value reveals that A's rows aren't fully independent in a 3-dimensional sense.
Who should use it
- Dimensionality reduction and principal component analysis.
- Determining a matrix's rank and finding a low-rank approximation of it.
Industry applications
- Machine learning and data science
- Image and signal processing
Advantages
- Works for any matrix, square or rectangular, unlike eigendecomposition.
- Directly reveals the matrix's rank via how many singular values are non-zero.
Limitations
- Results are numerical approximations, and very small non-zero singular values can be hard to distinguish from true zeros near the tool's numerical tolerance.
Common mistakes to avoid
- Confusing singular values with eigenvalues — for a non-symmetric matrix these are generally different numbers, though for a symmetric positive-definite matrix they coincide.
Best practices
- Verify your result by multiplying U·Σ·Vᵀ back together and confirming it reconstructs your original matrix A.
Tips
- The largest singular value gives you the matrix's 2-norm (its maximum "stretching factor" applied to any unit vector).