## Nielsen and Chuang Exercise 2.56

Exercise 2.56Use the spectral decomposition to show that is Hermitian for any unitary , and thus for some Hermitian .

In Exercise 2.18 we showed that the eigenvalues of a unitary matrix can be written for some real . Thus,

It then follows that

## Nielsen and Chuang Exercise 2.55

Exercise 2.55Prove that defined in Eq (2.91) is unitary.

Then,

Because is Hermitian, , and thus , since commutes with itself. By the previous exercise,

## Nielsen and Chuang Exercise 2.54

Exercise 2.54Suppose that and are commuting Hermitian operators. Verify that .

Because , we can find a simultaneous eigenbasis, such that and . Thus , and

## Nielsen and Chuang Exercise 2.53

Exercise 2.53What are the eigenvalues and eigenvectors of ?

There is no doubt a more elegant representation, but I worked these out numerically. The eigenvalues are , and the eigenvectors are given by the columns of

## Nielsen and Chuang Exercise 2.52

Exercise 2.52Verify that .

This was already proven in the previous example, since the Hadamard gate is Hermitian and unitary, it follows that .

## Nielsen and Chuang Exercise 2.51

Exercise 2.51Verify that the Hadamard gate is unitary.

In this case, since the Hadamard gate is Hermitian, , and thus we only have to prove that , since the other form . Thus,

## Nielsen and Chuang Exercise 2.50

Exercise 2.50Find the left and right polar decomposition of the matrix

For the left polar decomposition,

For the right polar decomposition

I worked out and from the spectral decomposition of and , respectively, and formed from .

## Nielsen and Chuang Exercise 2.49

Exercise 2.49Express the polar decomposition of a normal matrix in outer product representation.

A normal matrix is a matrix for which , and is guaranteed to have a spectral decomposition. A normal matrix also will have , so that the right and left polar decompositions are equal.

In outer product form,

where are the eigenvectors, and are the corresponding eigenvalues. Similarly,

leave a comment