Polar Decomposition

The polar decomposition of a square complex matrix A is a matrix decomposition of the form

where U is a unitary matrix and P is a positive-semidefinite Hermitian matrix. Intuitively, the polar decomposition separates A into a component that stretches the space along a set of orthogonal axes, represented by P, and a rotation represented by U. The decomposition of the complex conjugate of is given by .

This decomposition always exists; and so long as A is invertible, it is unique, with P positive-definite. Note that

gives the corresponding polar decomposition of the determinant of A, since and .

The matrix P is always unique, even if A is singular, and given by

where A* denotes the conjugate transpose of A. This expression is meaningful since a positive-semidefinite Hermitian matrix has a unique positive-semidefinite square root. If A is invertible, then the matrix U is given by

In terms of the singular value decomposition of A, A = W Σ V*, one has

confirming that P is positive-definite and U is unitary.

One can also decompose A in the form

Here U is the same as before and P′ is given by

This is known as the left polar decomposition, whereas the previous decomposition is known as the right polar decomposition. Left polar decomposition is also known as reverse polar decomposition.

The matrix A is normal if and only if P′ = P. Then UΣ = ΣU, and it is possible to diagonalise U with a unitary similarity matrix S that commutes with Σ, giving S U S* = Φ−1, where Φ is a diagonal unitary matrix of phases eiφ. Putting Q = V S*, one can then re-write the polar decomposition as

so A then thus also has a spectral decomposition

with complex eigenvalues such that ΛΛ* = Σ2 and a unitary matrix of complex eigenvectors Q.

The map from the general linear group GL(n,C) to the unitary group U(n) defined by mapping A onto its unitary piece U gives rise to a homotopy equivalence since the space of positive-definite matrices is contractible. In fact U(n) is a maximal compact subgroup of GL(n,C).