1. If . is an induced matrix norm, show that Show that the Frobenius norm is not an induced norm. 2. Let A be an matrix, its spectral radius, and be an induced matrix norm. Prove that for every...

1.

If . is an induced matrix norm, show that

Show that the Frobenius norm is not an induced norm.

2. Let A be an

matrix,

its spectral radius, and

be an induced matrix norm. Prove that for every

Hint: Begin by showing that

<sup>k

k</sup>

ifis an eigenvalue of
with corresponding eigenvector

3. Show that the Frobenius norm can be computed as

_F

­

<sup>T

1/2</sup>. or

_F

^T))
^1/2
.

4. Verify that

5. If A is an

matrix, prove that

6. Prove that if A is a symmetric matrix and

then

Use the spectral theorem.

7. Prove that if A is an
matrix and

then

Consult the hint for