## Nielsen and Chuang Exercise 2.20

Exercise 2.20Basis change: Suppose and are matrix representations of an operator on a vector space with respect to two different orthonormal bases, and . Then the elements of and are and . Characterize the relationship between and .

## Nielsen and Chuang Exercise 2.19

Exercise 2.19Show that the Pauli matrices are Hermetian and unitary.

Hermetian:

Since we’ve now shown that the matrices are Hermetian, it’s enough to show that the matrices square to :

## Nielsen and Chuang Exercise 2.18

Exercise 2.18Show that all eigenvalues of a unitary matrix have modulus 1, that is, can be written for some real .

Multiplying the two

or .

## Nielsen and Chuang Exercise 2.17

Exercise 2.17Show that a normal (i.e. diagonalizable) matrix is Hermitian if and only if it has real eigenvalues.

Part one: show that Hermiticity implies real eigenvalues. Since the matrix is diagonalizable, we can write

Thus

Since the matrix is Hermetian, , and thus , and thus the matrix has real eigenvalues.

Part two: Show that real eigenvalues imply Hermiticity. We may now write:

and following a similar procedure show that

Now, since the eigenvalues are real, , and thus , and thus the matrix is Hermitian.

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