Singular value decomposition

Every complex matrix has a factorization $A=V\Sigma U^{†}$ with $U,V$ unitary and $\Sigma$ diagonal with nonnegative entries. The squared singular values are the eigenvalues of $A^{†}A$.

Every complex symmetric matrix admits a unitary congruence decomposition with a unitary matrix $U$ and nonnegative diagonal entries $s_j$ equal to its singular values. This differs from unitary similarity diagonalization. It reduces a multimode squeezed vacuum to independent oscillator squeezing factors.

The singular values of a matrix $A$ are the nonnegative square roots of the eigenvalues of $A^{\ast }A$.

A unit right singular vector of a matrix or compact operator $A$ is an eigenvector of $A^{\ast }A$ with eigenvalue ${\sigma }_j^2$. For ${\sigma }_j>0$, $u_j=A{v}_j/{\sigma }_j$ is the corresponding left singular vector. The largest singular value is the operator norm of $A$, attained on its corresponding right singular vectors.

A unit left singular vector of a matrix or compact operator $A$ is an eigenvector of $A{A}^{\ast }$ with eigenvalue ${\sigma }_j^2$. For ${\sigma }_j>0$, it pairs with a right singular vector $v_j$ through $A{v}_j={\sigma }_j{u}_j$ and $A^{\ast }{u}_j={\sigma }_j{v}_j$.