SVD Calculator — Singular Value Decomposition A = UΣVᵀ of Any 2×2 to 4×4 Matrix

SVD Calculator

The singular value decomposition of matrices will never cause you any problems again — with the help of our SVD calculator, you will quickly master this important topic in linear algebra. Scroll down and learn:

• What the singular value decomposition is;
• How to find the SVD of a matrix using our SVD calculator;
• How to calculate the SVD of a matrix by hand; and
• Whether the singular value decomposition is unique.

Ready? Let's go!

What is the singular value decomposition?

Singular value decomposition (SVD) is a way of factorizing a matrix: any real matrix A of size m×n decomposes as

A = U Σ VT

where U and V are orthogonal matrices of sizes m×m and n×n respectively, and Σ (Sigma) is a rectangular matrix of the same size as A (so m×n) which has non-negative numbers on its diagonal and zeroes everywhere else. The diagonal elements of Σ are exactly the singular values of A.

The columns of U are called the left singular vectors, and the columns of V the right singular vectors. If A is complex, you replace the transpose V^T with the conjugate transpose V^*; then U and V become unitary matrices, but Σ still holds real non-negative numbers on its diagonal.

How to use this SVD calculator

Finding the SVD with our tool couldn't be simpler:

1. Pick the number of rows (m) and columns (n) of your matrix — anything from 2×2 up to 4×4.
2. Type your matrix entries into the grid on the left.
3. Read off the results on the right: the singular values σ₁ ≥ σ₂ ≥ …, the three factors U, Σ, and V^T, plus the matrix rank, the spectral norm, the Frobenius norm, and the condition number.

The calculator even verifies the result for you by reconstructing A = UΣV^T. If your matrix entries carry physical meaning, attach a metric, US customary, or currency context — 12 world currencies are supported, including the US dollar and the Russian ruble.

How to calculate SVD of a matrix by hand

The classic pen-and-paper route goes through the symmetric matrix A^TA:

1. Form A^TA (an n×n symmetric, positive semi-definite matrix).
2. Find its eigenvalues λ₁ ≥ λ₂ ≥ … ≥ 0. The singular values are their square roots: σ_i = √λ_i.
3. The right singular vectors (the columns of V) are the orthonormal eigenvectors of A^TA.
4. For each non-zero singular value, the matching left singular vector is u_i = A v_i / σ_i. Complete U to an orthonormal basis if any singular values are zero.
5. Assemble Σ with σ₁, σ₂, … on its diagonal.

Our calculator does all of this instantly, using a numerically stable one-sided Jacobi method so the result is accurate even when singular values repeat or vanish.

Is singular value decomposition unique?

Not entirely. The singular values themselves are unique — every matrix has exactly one set of them (up to ordering, and we always list them in decreasing order). The singular vectors, however, are not always unique:

• Each pair (u_i, v_i) tied to a non-repeated, non-zero singular value is determined up to a common sign flip.
• When a singular value is repeated, any orthonormal basis of its subspace works — so there are infinitely many valid U and V.
• The vectors corresponding to zero singular values can be any orthonormal completion of the basis.

So different tools (and different textbooks) may report U and V that look different yet are all correct — they always multiply back to the same A.

FAQs

What is the difference between SVD and eigendecomposition?
Eigendecomposition applies only to certain square matrices, while the SVD works for any matrix, square or rectangular. For a symmetric positive semi-definite matrix, the two coincide.

Can every matrix be decomposed with SVD?
Yes. Every real (or complex) matrix has a singular value decomposition — no exceptions.

What does the largest singular value tell me?
σ₁ is the spectral norm of the matrix: the largest factor by which the matrix can stretch a unit vector.

How is the rank related to the singular values?
The rank of the matrix equals the number of non-zero singular values.

What is the condition number?
In the 2-norm it is the ratio σ_max / σ_min. A large value means the matrix is ill-conditioned; if the smallest singular value is zero, the matrix is singular and the condition number is infinite.