Part I:

On the nal day of class, the following theorem was mentioned:

Theorem 0.1.

*Let A be an n×n self-adjoint matrix. Then there is a unitary matrix P such that P��*1*AP is*

diagonal. If, moreover,

A is a real symmetric matrix then there is an orthogonal matrix P such that P��

1*AP*

is diagonal.

The rst few problems will be concerned with making use of the full power of this theorem.

1. Determine an orthogonal matrix

*P *such that *P��*1*AP *is diagonal, where *A *is the 3 *× *3 matrix given

by

A

=

0

@

2

*−*1 0

−

1 2 *−*1

0

*−*1 2

1

A

*:*

2. Find a unitary matrix

*U *such that *U��*1*BU *is diagonal, where *B *is the 3 *× *3 matrix given by

B

=

0

@

2 2 1 +

*i*

2 2 1 +

*i*

1

*− i *1 *− i *1

1

A

*:*

One is not always so lucky as to have a self-adjoint matrix, however, and so the following more general

theorem is often useful.

Theorem 0.2.

*If A is an n × n normal matrix (i.e., if AA *= *AA) then there is a unitary matrix P such*

that

P��

1*AP is diagonal.*

3. Let

*C *be the 3 *× *3 matrix given by

C

=

0

@

0 1 + 2

*i *5*i*

−

1 + 2*i *0 2 + *i*

5

*i −*2 + *i *0

1

A

*:*

Test the matrix

*C *to determine whether or not *C *is normal, and if *C *should prove to be normal, nd a

unitary matrix

*U *such that *U��*1*CU *is a diagonal matrix.

1

4. Prove that any skew-adjoint

*n × n *matrix *A *(that is, *A *= *−A*) is normal, and thus diagonalizable.

(Note that the matrix

*C *of problem 3 above is skew-adjoint.)

More generally, we have the following:

Theorem 0.3

(Spectral Theorem)**. ***Let T be a normal operator on a nite-dimensional complex inner product*

space

V or a self-adjoint operator on a nite-dimensional real inner product space V . Let

1*; *2*; : : : ; k be*

the distinct eigenvalues of

T. Let Ej be the eigenspace associated with j , and let Pj be the orthogonal

projection of

V on Ej . Then Ei is orthogonal to Ej when i ̸

= *j, V *= *E*1 *⊕ E*2 *⊕ · · · ⊕ Ek, and*

T

= *1**P*1 + *2**P*2 + *· · · *+ *kPk:*

5. Suppose

*V *is an inner product space and that *W*1 and *W*2 are subspaces of *V *. Show that (*W*1+*W*2)*? *=

W

?

1

∩

W?

2

.

6. Suppose that

*V *is an inner product space, and let *W *be a subspace of *V *. Prove that

•

W ⊂ W??

, and

•

if *V *is nite-dimensional then *W *= *W??*.

7. The conclusion

*W *= *W?? *from the previous problem does not hold, in general, without the assump-

tion that

*V *is nite-dimensional. To see that this is so, do problem 23 of *§*6.2.

2

Subject | Mathematics |

Due By (Pacific Time) | 12/12/2012 |

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