How to use the svd calculator
- Enter the matrix, one row per line, with entries separated by spaces or commas. Up to 8 rows and 8 columns.
- Read the singular values, rank, condition number and the norms.
- Open the tables for the singular values and for the matrices U and V.
- Check the result by multiplying U, the diagonal of singular values and the transpose of V; you get the original matrix back.
Formula
A = U S V^T, where U and V have orthonormal columns and S is diagonal with s1 >= s2 >= ... >= 0
The singular values are the square roots of the eigenvalues of A^T A
Rank = the number of singular values above max(rows, columns) x s1 x 2.22e-16
Condition number = s1 / s(min); spectral norm = s1; Frobenius norm = sqrt(sum of s^2); nuclear norm = sum of s
The decomposition is found with one-sided Jacobi rotations, which are accurate even for small singular values. The signs of each pair of columns u and v can flip together.
Three small matrices
For the matrix with rows (3, 0) and (4, 5), the singular values are the square roots of 45 and 5, which are 6.7082 and 2.2361. The rank is 2, the condition number is 3, and the Frobenius norm is the square root of 50, about 7.0711. The matrix with rows (1, 2), (3, 4), (5, 6) has singular values 9.5255 and 0.5143, so its condition number is 18.5213.
The 3 by 3 matrix with rows (1, 2, 3), (2, 4, 6), (3, 6, 9) is the outer product of one column with one row, so it has rank 1. Its singular values are 14, 0 and 0, and the condition number is not defined. A wide 2 by 4 matrix with rows (2, 0, 1, 3) and (1, -1, 0, 2) has singular values 4.3525 and 1.0275.
What the singular values tell you
The singular values measure how much the matrix stretches space along perpendicular directions. The largest is the biggest stretch and the smallest is the weakest. A ratio of the two, the condition number, tells how much errors in the input can grow when you solve a system with this matrix. A very large condition number warns that the matrix is close to singular and answers will be sensitive.
The number of nonzero singular values is the rank. Data scientists keep only the largest few to compress a table or an image, because the best rank-k approximation of A is found by keeping the first k terms of U S V^T. This is the idea behind principal component analysis.
Related tools
For addition, multiplication, determinants and inverses with exact fractions, use the matrix calculator. If your matrix is symmetric and positive definite, the Cholesky decomposition calculator gives a faster factorization. To fit a line by least squares, which uses the same ideas, use the linear regression calculator.
For rotations in space, the rotation matrix and quaternion converter builds orthogonal matrices, which are exactly the matrices whose singular values are all 1.
Assumptions and limits
- Entries are real numbers. Complex matrices are not supported.
- The matrix is at most 8 by 8. Results are shown to 6 or 8 significant digits.
- Singular values below the rank tolerance are shown as 0.
- For a matrix with repeated singular values the columns of U and V are not unique, only the products are.
Frequently asked questions
What is singular value decomposition?
A way to write any matrix as a rotation, a stretch along the axes, and another rotation. The stretches are the singular values.
How is rank found?
By counting the singular values that are clearly above zero. Tiny values caused by rounding are treated as zero using the standard tolerance.
What does a large condition number mean?
The matrix is nearly singular. Small changes in the data can cause large changes in the solution of a linear system.
Why can the signs of U and V differ from another program?
Each pair of columns can be multiplied by -1 together without changing the product. This page makes the largest entry of each column of V positive.