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'''Singular values''' are '''Singular values''' are a generalization of [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]].
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Any linear transformation can be rewritten as a rotation (i.e., from the original basis onto a convenient basis), a scaling and stretching, and another rotation (i.e., from the convenient basis onto the destination basis). Therefore '''''A''' = '''UΣV'''^T^''. It is known that (for square matrices) there are [[LinearAlgebra/EigenvaluesAndEigenvectors|certain vectors]] which the transformation only scales: '''''A'''x = λx''.
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Alternatively, it is known that (for square matrices) there are [[LinearAlgebra/EigenvaluesAndEigenvectors|certain vectors]] which the transformation only scales: '''''A'''x = λx''.

For any matrix '''''A''''' of size ''m x n'', there are certain [[LinearAlgebra/Orthogonality|orthogonal]] [[LinearAlgebra/Basis|basis vectors]] ''u,,i,,'' in the column space that correspond to certain orthogonal basis vectors ''v,,i,,'' in the row space. They correspond such that '''''A'''v = σu''. And by composing these vectors and values together: '''''AV''' = '''UΣ'''''.		 Similarly, for any matrix '''''A''''' of size ''m x n'', there are certain vectors ''u,,i,,'' in the column space that correspond to certain vectors ''v,,i,,'' in the row space. They correspond such that '''''A'''v = σu''. And by composing these vectors and values together: '''''AV''' = '''UΣ'''''.
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Singular values are the square roots of [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. Specifically, a matrix '''''A''''' has singular values equal to the square roots of the eigenvalues of '''''A'''^T^'''A''''': ''σ = √λ''. Singular values are square roots of [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. Specifically, a matrix '''''A''''' has singular values equal to the square roots of the eigenvalues of '''''A'''^T^'''A''''': ''σ = √λ''.
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Matrices of all sizes have singular values. For a matrix of size ''m x n'', there are at most ''min{m,n}'' singular values. Furthermore, the number of non-zero singular values is equal to the [[LinearAlgebra/Rank|rank]] of '''''A'''''.

These singular values are the numbers on the diagonal of '''''Σ'''''. Naturally '''''Σ''''' matches '''''A''''' as a ''m x n'' matrix.		 Matrices of all sizes have singular values. The number of non-zero singular values is equal to the [[LinearAlgebra/Rank|rank]] of '''''A'''''.

Singular Values

Singular values are a generalization of eigenvalues.

Contents

  1. Singular Values
    1. Introduction
    2. Description

Introduction

It is known that (for square matrices) there are certain vectors which the transformation only scales: Ax = λx.

Similarly, for any matrix A of size m x n, there are certain vectors ui in the column space that correspond to certain vectors vi in the row space. They correspond such that Av = σu. And by composing these vectors and values together: AV = UΣ.

Description

Singular values are square roots of eigenvalues. Specifically, a matrix A has singular values equal to the square roots of the eigenvalues of ATA: σ = √λ.

Matrices of all sizes have singular values. The number of non-zero singular values is equal to the rank of A.

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LinearAlgebra/SingularValues (last edited 2026-02-08 00:19:22 by DominicRicottone)