UCLA Linear algebra Polar Decomposition Problems Homework


Linear algebra polar decomposition. All details are in the picture. I wanna know the exact steps to do a) b) c)

3. (25 points) Polar Decomposition
Let A be an invertible linear operator on a finite-dimensional complex vector space V. Recall
that we have shown in class that in this case, there exists a unique unitary operator U such that
A = U|A|. The point of this exercise is to prove the following result: an invertible operator A
is normal if and only if U|A| = |A|U.
a) Show that if U|A| = |A|U, then AA* = A* A.
Now, we want to show the other direction, i.e. if AA* = A*A, then U|A| = |A|U, which is going
to be more difficult.
b) Show that if A is normal, then U|A|= |AU.
c) We now want to finish the proof by concluding that U|A|2 = |A|PU implies U|A| = |A|U.
For notational convenience, define B := |A|2 and use the spectral theorem to show that there
exists a polynomial g(t) such that g(B) = VB. Use this to conclude U|A| = |A|U.

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