exploit the spectral decomposition of to write . In this expression, is unitary because is. To show that also is unitary, we can use the SVD to write , so that where again is unitary by construction. Yet another way to directly show the unitarity of is to note that, writing the SVD of in terms of rank-1 matrices as , where are the singular values of , we have which directly implies the unitarity of because a matrix is unitary if and only if its singular values have unitary absolute value. Note how, from the above construction, it follows that the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined. General derivation The SVD of a squared matrix reads , with unitary matrices, and a diagonal, positive semi-definite matrix. By simply inserting an additional pair of s or s, we obtain the two forms of the polar decomposition of :More generally, if is some rectangular matrix, its SVD can be written as where now and are isometries with dimensions and , respectively, where , and is again a diagonal positive semi-definite squared matrix with dimensions . We can now apply the same reasoning used in the above equation to write , but now is not in general unitary. Nonetheless, has the same support and range as , and it satisfies and . This makes into an isometry when its action is restricted onto the support of , that is, it means that is a partial isometry. As an explicit example of this more general case, consider the SVD of the following matrix:We then havewhich is an isometry, but not unitary. On the other hand, if we consider the decomposition ofwe findwhich is a partial isometry (but not an isometry). Bounded operators on Hilbert space The polar decomposition of any bounded linear operator A between complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative operator. The polar decomposition for matrices generalizes as follows: if A is a bounded linear operator then there is a unique factorization of A as a product A = UP where U is a partial isometry, P is a non-negative self-adjoint operator and the initial space of U is the closure of the range of P. The operator U must be weakened to a partial isometry, rather than unitary, because of the following issues.