Josalynn

2021-02-08

Let A be a $nxxn$ matrix and let
$B=A+{A}^{T}andC=A-{A}^{T}$
(a) Show that B is symmetric and C is skew symmetric.
(b) Show that every n × n matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.

Nicole Conner

Expert

as we know that a matrix E is said to be symmetric if ${E}^{T}=E$ and skew symmetric if ${E}^{T}=-E.$
as $B=A+{A}^{T}$ therefore,
$={B}^{T}=\left(A+{A}^{T}{\right)}^{T}$
$={A}^{T}+\left({A}^{T}{\right)}^{T}$
$={A}^{T}+A\left(As\left({A}^{T}{\right)}^{T}=A\right)$
=B
as ${B}^{T}=B$ therefore, B is symmetric matrix.
hence proved.
As C=A−AT
therefore, $CT=\left(A-{A}^{T}{\right)}^{T}$
$={A}^{T}-\left({A}^{T}\right)T$
$={A}^{T}-A$
$=-\left(A-{A}^{T}\right)$
=−C
as ${C}^{T}=-C$ therefore, C is skew symmetric matrix. Hence proved.
Now we have to show that every nxxn matrix can be expressed as the sum of the symmetric and skew symmetric matrix. Let A be nxxn matrix. therefore,
$A=1/2\left(2A\right)$
$=1/2\left(A+A\right)$
$=1/2\left(A+{A}^{T}+A-{A}^{T}\right)$
$=1/2\left(B+C\right)$
where $B=A+{A}^{T}$ and B is a symmetric matrix and $C=A-{A}^{T}$ and C is a skew symmetric matrix.
herefore,
$A=1/2\left(B+C\right)$
$=1/2$(symmetric matrix +skew symmetric matrix)
therefore, it has been showed that any matrix A of order nxxn can be expressed as the sum of symmetric and skew symmetric matrix.

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