Project #1605 - Linear algeba

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��1AP is

 

diagonal. If, moreover,

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

1AP

 

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��1AP 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��1BU 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��

1AP is diagonal.

 

3. Let

C be the 3 × 3 matrix given by

 

C

=

 

0

@

0 1 + 2

i 5i

 

1 + 2i 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��1CU 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 = E1 E2 ⊕ · · · ⊕ Ek, and

 

T

= 1P1 + 2P2 + · · · + kPk:

 

5. Suppose

V is an inner product space and that W1 and W2 are subspaces of V . Show that (W1+W2)? =

 

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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