Perform the polar decomposition of a 2×2 matrix:

The matrix is positive definite and real symmetric:

The matrix is real orthogonal:

The product is the original matrix:

Compute the polar decomposition for a 4×3 matrix :

The matrix has the same shape as , while is square:

Confirm the decomposition:

Since is rectangular, being real (semi-)orthogonal means PseudoInverse[u] == Transpose[u]:

The matrix is real symmetric and positive semidefinite:

Compute the polar decomposition of a complex matrix :

As is complex valued, is a complex-valued Hermitian matrix:

Despite being complex valued, is still positive semidefinite:

Moreover, is a complex-valued unitary matrix:

Confirm the decomposition up to numerical error:

A matrix with entries having 20-digit precision:

The polar decomposition is computed using the full 20-digit precision of the input:

Check the result:

The polar decomposition of an exact matrix:

As expected, is square, real symmetric and positive semidefinite:

And is a real orthogonal matrix of the same shape as :

Verify the decomposition:

Consider the polar decomposition of the following noninvertible matrix:

The matrix is uniquely defined by the requirement that it equal the non-negative matrix square root, :

However, only has to satisfy the conditions and (since is real valued). Solve these equations for a general matrix, where the latter equation has been rewritten as :

The first solution is the matrix returned by PolarDecomposition:

The second solution is a different unitary matrix:

Confirm that is also a unitary matrix that reconstructs :

For a matrix, PolarDecomposition[m] equals {Exp[I Arg[m]],Abs[m]}:

For Hermitian positive semidefinite input h, the positive symmetric part of the decomposition is h itself:

The unitary part of the decomposition is simply an identity matrix:

For unitary m, the unitary part of PolarDecomposition[m] is m, itself:

The positive definite Hermitian part is an identity matrix:

If {u,p}==PolarDecomposition[m], then p==MatrixPower[ConjugateTranspose[m].m, 1/2]:

And u satisfies ConjugateTranspose[u]==PseudoInverse[u]:

If {u,p}==PolarDecomposition[m], then u and m have the same dimensions:

p is a square matrix with the same number of columns as m:

If the input is real, PolarDecomposition returns OrthogonalMatrix and SymmetricMatrix objects:

Otherwise it returns UnitaryMatrix and HermitianMatrix objects:

A matrix is normal if and only if the matrices in its polar decomposition commute. First, consider a normal matrix:

The pieces of its polar decomposition commute ():

Next, consider a non-normal matrix:

The pieces of its polar decomposition do not commute:

A polar decomposition can be recovered from the singular value decomposition as and :

QRDecomposition factorizes a matrix into unitary and upper triangular matrices:

PolarDecomposition, by contrast, factorizes into unitary and Hermitian positive semidefinite matrices:

PolarDecomposition can give an unwieldy result for exact matrices:

For even modest dimensions, the polar decomposition of an exact matrix can be quite slow: