## Nielsen and Chuang Exercise 2.26

Exercise 2.26Let . Write out and explicitly, both in terms of tensor products like , and using the Kronecker product.

In terms of tensor products, . Writing this as a Kronecker product yields:

Similarly, . Writing this as a Kronecker product yields:

## Nielsen and Chuang Exercise 2.25

Exercise 2.25Show that for any operator , is positive.

Because , trivially, and thus is normal, and thus has an eigendecomposition. We can therefore write:

Multiplying both sides on the left by :

Defining yields

thus any eigenvalue may be taken to be nonnegative, and thus the matrix is positive.

## Nielsen and Chuang Exercise 2.24

Exercise 24Show that a positive operator is necessarily Hermitian. Hint: show that an arbitrary operator can be written , where and are Hermitian.

I don’t really get the point of this—obviously if a matrix has all nonnegative eigenvalues they are trivially real, and the matrix is Hermitian. However, I think this is something along the lines of what they want:

By defining the matrices and we can write where and is Hermitian. We can then show that the eigenvalues of are given by:

where are real. Thus the eigenvalues of can be written as the sum of . If the matrix is positive, the imaginary part must be zero, so that the eigenvalues are all real and the matrix is Hermitian

## Nielsen and Chuang Exercise 2.23

Exercise 2.23Show that the eigenvalues of a projector are all either 0 or 1.

We can write a projector as

We may also define the projector of the orthogonal space in a diagonal basis (exercise 2.22)

and can thus expand in an eigenbasis with either eigenvalue 1 (for the space), or eigenvalue 0 (for the space)

and thus the eigenvalues are either 0 or 1.

## Nielsen and Chuang Exercise 2.22

Exercise 2.22Prove that two eigenvectors of a Hermitian operator with different eigenvalues are necessarily orthogonal.

A Hermitian operator can act to the right or the left. Thus, if we have

then

(acting to the right), or

(acting to the left). Subtracting the two:

so that either or

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