QMISMF Chap 3 « Dan's Thoughts

21Oct/090

QMISMF Chap 3

3-1
$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}+\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}=\begin{pmatrix} 1 & -i \\ i & -1 \end{pmatrix}$
$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}+\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}=\begin{pmatrix} 0 & 1-i \\ 1+i & 0 \end{pmatrix}$
$\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}+\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}=\frac{1}{2}\begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}+\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}=\begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix}$
3-2
$\begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & 0 \\ 2 & 0 & 3 \end{pmatrix} \begin{pmatrix} 3 & 3 & -1 \\ 0 & 1 & 0 \\ -2 & -2 & 1 \end{pmatrix}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
$\begin{pmatrix} 3 & 3 & -1 \\ 0 & 1 & 0 \\ -2 & -2 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & 0 \\ 2 & 0 & 3 \end{pmatrix} =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
3-3
$\frac{1}{4}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}=\frac{1}{4}\begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix}=\frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$
$\frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}\frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}=\frac{1}{4}\begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}=\frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$
$\frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}=\frac{1}{4}\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$
$\frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}\frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}=\frac{1}{4}\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$
3-4
$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{pmatrix} \begin{pmatrix} 1 & -3 & 2 \\ -3 & 3 & -1 \\ 2 & -1 & 0 \end{pmatrix} =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
3-5
Assume: $AM^{-1}=M^{-1}A \iff AM=MA$
$AM^{-1}=M^{-1}A \implies AM=MA$
$AM^{-1}=M^{-1}A$
$AM^{-1}M=M^{-1}AM$
$MAI=MM^{-1}AM$
$MA=IAM$
$MA=AM$

$AM=MA \implies AM^{-1}=M^{-1}A$
$AM=MA$
$M^{-1}AM=A$
$M^{-1}A=AM^{-1}$
3-6
Assume: $AM=MA$ and $BM=MB$
$(A+B)M=M(A+B)$
$AM+BM=MA+MB$
$AM+BM-MA-MB=0$
$(AM-MA)+(BM-MB)=0$
$0+0=0$ $(AB)M=M(AB)$
$A(BM)=(MA)B$
$A(MB)=(AM)B$
$(AM)B=(AM)B$ $(zB)M=M(zB)$
$z(BM)=(MB)z$
$z(BM)=(BM)z$

3-7
$\begin{pmatrix} 2 & 2-i \\ 2+i & -2 \end{pmatrix} \frac{1}{9}\begin{pmatrix} 2 & 2-i \\ 2+i & -2 \end{pmatrix}=\frac{1}{9}\begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

Dan's Thoughts Thinking somewhat carefully

QMISMF Chap 3

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