Question

$Express \left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right] as a sum of a symmetric matrix and a skew-symmetric matrix.$

Answer 1

HELLO DEAR,


let $\bold{A = \left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right]}$



=> A' = $\left[\begin{array}{ccc}1&2&4\\-1&1&-1\\2&3&5\end{array}\right]$



therefore, (A + A') = $\bold{\left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right]}$ + $\left[\begin{array}{ccc}1&2&4\\-1&1&-1\\2&3&5\end{array}\right]$

= ${\left[\begin{array}{ccc}(1 + 1)&(-1 + 2)&(2 + 4)\\(2 - 1)&(1 + 1)&(3 - 1)\\(4 + 2)&(3 - 1)&(5 + 5)\end{array}\right]}$

= $\left[\begin{array}{ccc}2&1&6\\1&2&2\\6&2&10\end{array}\right]$



let p = 1/2(A + A') = $\left[\begin{array}{ccc}1&(1/2)&3\\(1/2)&1&1\\3&1&5\end{array}\right]$


AND,

(A - A') = $\bold{\left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right]}$ - $\left[\begin{array}{ccc}1&2&4\\-1&1&-1\\2&3&5\end{array}\right]$


= $\left[\begin{array}{ccc}(1 - 1)&(-1 - 2)&(2 - 2)\\(2 + 1)&(1 - 1)&(3 + 1)\\(4 - 2)&(-1 - 3)&(5 - 5)\end{array}\right]$

let Q = 1/2(A - A') = 1/2.$\left[\begin{array}{ccc}0&-3&-1\\3&0&4\\2&-4&0\end{array}\right]$ = $\left[\begin{array}{ccc}0&(-3/2)&(-1/2)\\(3/2)&0&2\\1&-2&0\end{array}\right]$



NOW,

p' = $\left[\begin{array}{ccc}1&(1/2)&3\\(1/2)&1&1\\3&1&5\end{array}\right]'$ = $\left[\begin{array}{ccc}1&(1/2)&3\\(1/2)&1&1\\3&1&5\end{array}\right]$ = p

therefore, p is symmetric.


AND Q' = $\left[\begin{array}{ccc}0&(-3/2)&(-1/2)\\(3/2)&0&2\\1&-2&0\end{array}\right]'$ = $\left[\begin{array}{ccc}0&(-3/2)&(-1/2)\\(3/2)&0&2\\1&-2&0\end{array}\right]$ = -Q.

therefore, Q is skew-symmetric.


NOW, (P + Q) =( $\left[\begin{array}{ccc}1&(1/2)&3\\(1/2)&1&1\\3&1&5\end{array}\right]$ + $\left[\begin{array}{ccc}0&(-3/2)&(-1/2)\\(3/2)&0&2\\1&-2&0\end{array}\right]$)

= $\left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right]$ = A.

hence, A = P + Q.

where, p is symmetric and Q is skew-symmetric.


I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

Hello,

Solution:

$\left[\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right]$

To express given matrix in sum of symmetric and skew symmetric matrix

First write the sum of matrix and it's transpose
[ A+ A']=
$\left [\begin{array}{ccc}1&-1&2\\2&1&3\\4&-1&5\end{array}\right] \: + \left [\begin{array}{ccc}1&2&4\\-1&1&-1\\2&3&5\end{array}\right]\\ \\ A + A' = \left[\begin{array}{ccc}2&1&6\\1&2&2\\6&2&10\end{array}\right]\\\\ let P = 1/2 [ A +A']\\ \\ = \left[\begin{array}{ccc}1&1/2 &3\\1/2&1& 1\\3&1&5\end{array}\right]\\ \\ A - A' = \left[\begin{array}{ccc}0&-3&-2\\3&0&4\\2&-4&0\end{array}\right]\\ \\ Let Q = 1/2 [A - A']\\\ \\ =\left[\begin{array}{ccc}0&-3/2&-1\\3/2&0&2\\1&-2&0\end{array}\right] \\ \\ A \:\: in\:\: the\:\: form\:\: \\ \\of \:\:sum \:\:of \:\:symmetric \:\:and \:skew symmetric\:\: matrix\\ \\ P+Q= \left[\begin{array}{ccc}1&1/2 &3\\1/2&1& 1\\3&1&5\end{array}\right]+ \left[\begin{array}{ccc}0&-3/2&-1\\3/2&0&2\\1&-2&0\end{array}\right]$
here P is symmetric matrix and Q is skew -symmetric matrix.

hope it helps you.