Singular Values Calculator (SVD)

Compute singular values of a 2x2 or 3x3 matrix using Singular Value Decomposition.

Enter Matrix

Enter matrix entries row by row:

Result

Singular Values
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σ values

Step-by-Step Solution

Understanding Singular Value Decomposition

Singular Value Decomposition (SVD) is a fundamental matrix factorization technique in linear algebra. For any m x n matrix A, SVD decomposes it into three matrices: A = U Σ VT, where U and V are orthogonal matrices and Σ is a diagonal matrix containing the singular values.

How SVD Works

The Decomposition

Any matrix A can be written as A = UΣVT where U and V are orthogonal and Σ is diagonal.

A = U Σ V^T

Finding Singular Values

Singular values are the square roots of the eigenvalues of ATA (or AAT).

σ_i = sqrt(eigenvalue_i(A^T A))

Properties

Singular values are always non-negative and conventionally ordered from largest to smallest.

σ_1 ≥ σ_2 ≥ ... ≥ 0

Applications of SVD

SVD is widely used in data science for dimensionality reduction (PCA), image compression, recommendation systems, solving least-squares problems, computing matrix pseudoinverses, and determining matrix rank. It is one of the most important decompositions in numerical linear algebra.

Key Facts

  • The number of non-zero singular values equals the rank of the matrix.
  • The largest singular value is the matrix 2-norm (spectral norm).
  • The ratio of the largest to smallest singular value gives the condition number.
  • SVD always exists for any real or complex matrix.