Singular value

A

are the nonnegative square roots of the eigenvalues of

A^{*} A

Right singular vector ( $v_{j}$ )

A

A^{*} A

σ_{j}^{2}

. For

σ_{j} > 0

u_{j} = A v_{j} / σ_{j}

is the corresponding left singular vector. The largest singular value is the operator norm of

A

, attained on its corresponding right singular vectors.

A

A A^{*}

σ_{j}^{2}

. For

σ_{j} > 0

v_{j}

through

A v_{j} = σ_{j} u_{j}

and

A^{*} u_{j} = σ_{j} v_{j}

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Singular values are a set of values that arise from the singular value decomposition (SVD) of a matrix. The SVD is a fundamental technique in linear algebra and statistics that is used to factorize a matrix into three other matrices.