Solving a Linear Algebra Problem: Finding Coefficient and Orthogonal Matrices

[degs2k4]

Homework Statement

The Attempt at a Solution

on P1), they only ask for the coefficient matrix, which I think is the following one:
1 -6 2
0 1 4
0 0 15/8

on P2), they ask for the orthogonal matrix P used for the transformation. I suppose I have to apply Gram-Schmidt of the coefficient matrix from P1) to get P, but I am not sure of it...
Could someone guide me to solve this problem ?

Thanks in advance

 

[lanedance]

that matrix gives the quadratic form, but the normal way to do this is to consider the symmetric matrix that gives your equation, so if we call your matrix M, we can find the symmetric part by:
[tex] A = \frac{1}{2}(M + M^T)
= \frac{1}{2}(\begin{pmatrix} 1 & -6 & 2 \\0 & 1 & 4 \\0 & 0 & \frac{15}{8} \end {pmatrix}
+ \begin{pmatrix} 1 & 0 & 0 \\-6 & 1 & 0 \\2 & 4 & \frac{15}{8} \end {pmatrix})
= \begin{pmatrix} 1 & -3 & 1 \\-3 & 1 & 2 \\1 & 2 & \frac{15}{8} \end {pmatrix}
[/tex]

you can check this gives you the same quadratic form, and the properties of the symmetric matrix will be useful later on

 

[lanedance]

for the next parts, i would start thinking about eignevectors...

note that you can prove symmetric matricies with real entries have real eigenvalues & are diagonalisable

 

[degs2k4]

that matrix gives the quadratic form, but the normal way to do this is to consider the symmetric matrix that gives your equation...

Why? I mean, that is something you thought of by looking at the rest of the problem? I didn't know that was the normal way to do that...

for the next parts, i would start thinking about eignevectors...
note that you can prove symmetric matricies with real entries have real eigenvalues & are diagonalisable

Thanks for the advice. I'm going to try that and post back the results here again.

 

[lanedance]

Why? I mean, that is something you thought of by looking at the rest of the problem? I didn't know that was the normal way to do that...

i had a look at my textbook... ;)

the explanation is as follows, first any matrix can be written in terms of its symmetric (A=A^T) and anti symmetric (B=-B^T) parts, say M = A + B, where they are given by:
[tex]
A = \frac{1}{2}(M+ M^T), \ \ B = \frac{1}{2}(M- M^T)
[/tex]

now consider the quadratic form
[tex]
Q = x^T A x
[/tex]

now consider the quadratic form
[tex]
Q = x^T M x = x^T A +x ^T B x
[/tex]

Q is scalar, so clearly Q^T = Q, then
[tex]
Q = x^T M x = x^T A +x ^T B x = Q^T = x^TA x - x ^T B x
[/tex]

which gives
[tex]
x ^T B x = 0
[/tex]

so the anti symmetric part doesn't really do anything anyway..

 

[lanedance]

then when you do it, you get all of the nice properties that come with symmetric matricies

 

[degs2k4]

Hello again,

Sorry about the late reply. Have been quite busy with other work.

After learning the properties of symmetric matrices and eigenvalues/vectors, I think I could solve most of the problem. But have doubts in the last parts of the problem. My main doubt now is how could I express the translation of section (3) as a matrix to be used in (4) for the composite transformation...

I uploaded 4 scanned pages of what I have done so far:

http://i50.tinypic.com/zsvlmg.jpg"
http://i49.tinypic.com/1zyydkh.jpg"
http://i45.tinypic.com/2mynbqb.jpg"
http://i47.tinypic.com/bjipmc.jpg"

I would be very grateful if someone could check it...
Thanks in advance...

 

[degs2k4]

Could some please check if this is ok, specially parts 3-5?

Thanks

Page 1 : sections (1) and (2) of the problem

Page 2 : section (2) (cont)

Page 3 : section (3)

Page 4 : section (4) and (5)