# Singular value decomposition

From AMS Glossary

## singular value decomposition

A numerical procedure to bring a matrix on diagonal form with its singular values on the diagonal.

For a given matrix 𝗠 the method finds orthogonal matrices 𝗨 = [

**u**_{1}, . . . ,**u**_{n}] and 𝗩 = [**v**_{1}, . . . ,**v**_{n}] so that 𝗨^{T}𝗠𝗩 = Σ with Σ =*diag*(σ_{1}, . . . , σ_{n}). The columns of 𝗨 and 𝗩 are the left and right singular vectors of 𝗠, respectively, the σ_{i}are the singular values, and 𝗠**u**_{i}= σ_{i}**v**_{i}.