Question

i. Let A= [ 5 0 0]
[ 1 5 0]
[ 0 1 5]
Find a column vector X for which A= cX, for some c belong to R.

ii. Give an example with justification, of a skew-hermitian operator on C^2

Answer 1

$A=\begin{vmatrix} 5&0&0 \\ 1&5&0 \\0&1&5 \end{vmatrix}$

you cannot find the column X such that A=cX for c$\in$ R

since A is 3x3 and cX is 3x1

however, we can find X such that AX=cX for some c$\in$ A

clearly, 5 is an eign calue & A and the compounding eigen vector is

$X=\begin{bmatrix} 0\\0\\a \end{bmatrix}$

foe any a$\in$ R

we can check that AX=5X

$AX=\begin{bmatrix} 5 & 0&0 \\1&5&0\\0&1&5 \end{bmatrix}$ $\begin{bmatrix} 0\\0\\a \end{bmatrix}$ =$X=\begin{bmatrix} 0\\0\\5a \end{bmatrix}$ =$5\begin{bmatrix} 0\\0\\a \end{bmatrix}$ =$5X$

as,$A=\begin{pmatrix} -1^0 & 2+1^0 \\ -2+1^0 & 0 \end{pmatrix}$

$A^T=\begin{bmatrix} -1^0&-2+1^0\\2+1^0&0 \end{bmatrix}$

$A^T conjugate =(A^T)^*=\begin{bmatrix} 1^0&-2+1^0\\2-1^0&0 \end{bmatrix}$ $=$ $=\begin{bmatrix} -1^0&2+1^0\\-2-1^0&0 \end{bmatrix}$

$\therefore$ conjugate transpose of A is -A

Therefpre A is skew-hermitian