Linear Algebra - Polar decomposition

[Mumba]

First i calculated the eigenvalues: I got
$$(i-\lambda)(-i-\lambda)+1$$, so
$$\lambda_{1,2}=+-\sqrt{2}i$$

Is it correct to go on on like this:

$$\lambda_{1}a+b=\sqrt{\lambda_{1}}$$
$$\lambda_{2}a+b=\sqrt{\lambda_{2}}$$

After calculating a and b, we plug it into f(x) = ax+b -->
$$f(A^{*}A)=a(A^{*}A)+bI$$

Then
$$f(A^{*}A)=\sqrt{A^{*}A}=|A|$$ and
$$U=A|A|^{-1}$$

This way i find U, and i think |A|=P

So i have the polar decompostion A = UP?!
Is the way correct?

Thx
Mumba

Edit: A* - Transpose

 

[lippyka]

of matrix AYes, your approach is correct. The polar decomposition of a matrix A is given by A = UP, where U is a unitary matrix and P is a positive semidefinite matrix. To find U, you need to calculate the eigenvalues of A*A, then take their square roots, and then solve the equation (A*A - λI)x = 0 for x to find the eigenvectors. Then construct U from the eigenvectors, and P from the eigenvalues.