Editing Polar decomposition (section)

=== Derivation for invertible matrices ===
From the [[singular-value decomposition]], it can be shown that a matrix <math>A</math> is invertible if and only if <math>A^* A</math> (equivalently, <math>AA^*</math>) is. Moreover, this is true if and only if the eigenvalues of <math>A^* A</math> are all not zero.<ref>Note how this implies, by the positivity of <math>A^* A</math>, that the eigenvalues are all real and strictly positive.</ref>

In this case, the polar decomposition is directly obtained by writing
<math display="block">A = A\left(A^* A\right)^{-1/2}\left(A^* A\right)^{1/2},</math>
and observing that <math>A\left(A^* A\right)^{-1/2}</math> is unitary. To see this, we can exploit the spectral decomposition of <math>A^* A</math> to write <math>A\left(A^* A\right)^{-1/2} = AVD^{-1/2}V^*</math>.

In this expression, <math>V^*</math> is unitary because <math>V</math> is. To show that also <math>AVD^{-1/2}</math> is unitary, we can use the [[singular-value decomposition|SVD]] to write <math>A = WD^{1/2}V^*</math>, so that 
<math display="block">AV D^{-1/2} = WD^{1/2}V^* VD^{-1/2} = W,</math>
where again <math>W</math> is unitary by construction.

Yet another way to directly show the unitarity of <math>A\left(A^* A\right)^{-1/2}</math> is to note that, writing the [[singular-value decomposition|SVD]] of <math>A</math> in terms of rank-1 matrices as <math display="inline">A = \sum_k s_k v_k w_k^*</math>, where <math>s_k</math>are the singular values of <math>A</math>, we have 
<math display="block">A\left(A^* A\right)^{-1/2}
= \left(\sum_j \lambda_j v_j w_j^*\right)\left(\sum_k |\lambda_k|^{-1} w_k w_k^*\right)
= \sum_k \frac{\lambda_k}{|\lambda_k|} v_k w_k^*,</math> 
which directly implies the unitarity of <math>A\left(A^* A\right)^{-1/2}</math> 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''.