What is SVD?

SVD is a matrix factorization method which decomposes a matrix into three other matrices.

  • U: An orthogonal matrix whose columns are called left-singular vectors of .
  • Σ: An diagonal matrix (though not necessarily square), whose non-negative diagonal elements are known as singular values of . The singular values are usually arranged in descending order along the diagonal.
  • V*: The conjugate transpose of an n×n orthogonal matrix whose columns are called right-singular vectors of .

Procedure Generalized

  1. Transpose the matrix and multiply:
  2. Compute eigenvalues:
  3. Substitute eigenvalues to compute eigenvectors
  4. Normalize each eigenvector and use the normalized columns to construct
  5. Construct from the square-root of the eigenvalues along the diagonals.
  6. Construct U from the process of

Examples from Textbook

(David C. Lay, Steven R. Lay, Judi J. McDonald-Linear Algebra and Its Applications-Pearson Education (c2016))


Construct a singular value decomposition of

is free


is free


Construct :

Construct V:

Compute :