(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use the Gram-Schmidt method to find an orthogonal matrix Q and an upper triangular matrix R so that A=QR, where

[tex]A=\begin{bmatrix}16 & -4 & 8\\-4 & 5 & -4\\8 & -4 & 14\end{bmatrix}[/tex]

2. Relevant equations

[tex]A^{(1)}=Q^{(1)}r_{11}[/tex]

[tex]A^{(2)}=Q^{(1)}r_{12}+Q^{(2)}r_{22}[/tex]

[tex]A^{(3)}=Q^{(1)}r_{13}+Q^{(2)}r_{23}+Q^{(3)}r_{33}[/tex]

[tex]r_{11}=||A^{(1)}||[/tex]

[tex]r_{12}=(Q^{(1)})^H A^{(2)}[/tex]

[tex]r_{22}=}||A^{(2)}-r_{12}Q^{(1)}||[/tex]

Those are the formulas we derived form the information in our book.

The [tex]A^{(1)}[/tex] represents the first column of the matrix A, and the same goes for the Q's. The [tex]r_{11}[/tex] is the 1,1 entry in the R matrix and so forth. The H is Hermetian.

3. The attempt at a solution

OK.

In order to find the r11 entry in my R matrix, I took the norm of the first column of A:

[tex]r_{11}=\sqrt{16^2+-4^2+8^2}=4\sqrt{21}[/tex]

Then I divide the first column of A by [tex]4\sqrt{21}[/tex] to gert my first column of Q.

That gives [tex]Q^{(1)}=

\left( \begin{matrix}\frac{4}{\sqrt{21}}\\ \frac{-1}{\sqrt{21}}\\ \frac{2}{\sqrt{21}}\end{matrix}\right)[/tex]

Then I need my r12 entry of my R matrix. So,

[tex]r_{12}=(Q^{(1)})^HA^{(2)}[/tex]

which is:

[tex]\left( \begin{matrix}\frac{4}{\sqrt{21}}&\frac{-1}{\sqrt{21}}& \frac{2}{\sqrt{21}}\end{matrix}\right)

\left( \begin{matrix}-4\\5\\-4\end{matrix}\right) =\frac{-29}{\sqrt{21}}[/tex]

Then I need my second column of my Q matrix, which is supposed to be given by

[tex]r^{22}Q^{(2)}=A^{(2)}-r_{12}Q^{(1)}[/tex]

according to the equation. I just solved it for r22Q2.

Anyway, this is where my trouble starts.

I got:

[tex]\left(\begin{matrix}-4\\5\\-4\end{matrix}\right)

- (-\frac{29}{\sqrt{21}})

\left(\begin{matrix}\frac{4}{\sqrt{21}}\\ \frac{-1}{\sqrt{21}}\\ \frac{2}{\sqrt{21}}\end{matrix}\right) =

\left( \begin{matrix}\frac{-200}{24}\\ \frac{134}{21}\\ \frac{-142}{21}\end{matrix}\right)[/tex]

Which is totally wrong. I know I'm missing something, but I can't see it.

The norm of that thing is SUPPOSED to be 1, but it's not.

I hoipe my latex came out OK. Bear with me while I try to edit it.

CC

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# Homework Help: Gram-Schmidt Orthogonalizationh

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