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  • Express as a sum of a symmetric and a skew symmetric m

Express A=\begin{bmatrix} 4 &-3 \\ 2& -1 \end{bmatrix} as a sum of a symmetric and a skew symmetric matrix.

 

 
 
 
 
 

Answers (1)

A=\begin{bmatrix} 4 &-3 \\ 2& -1 \end{bmatrix}

A^{T}=\left[\begin{array}{cc} 4 & 2 \\ -3 & -1 \end{array}\right] \\

P=\frac{A+A^{T}}{2}=\frac{1}{2}\left[\begin{array}{cc} 8 & -1 \\ -1 & -2 \end{array}\right] \\

P'=\frac{1}{2}\left[\begin{array}{cc} 8 & -1 \\ -1 & -2 \end{array}\right]=P\\

P\rightarrow $Symmetric matrix $\\

Q=\frac{A-A^{T}}{2}=\frac{1}{2}\left[\begin{array}{cc} 0 & -5 \\ 5 & 0 \end{array}\right] \\

Q'=\frac{1}{2}\left[\begin{array}{cc} 0 & 5 \\ -5 & 0 \end{array}\right] = -Q\\

Q\rightarrow $Skew -symmetric matrix $\\

\\ P+Q=\frac{1}{2}\left[\begin{array}{ll} 8 & -6 \\ 4 & -2 \end{array}\right]\\\\P+Q=\left[\begin{array}{cc} 4 & -3 \\ 2 & -1 \end{array}\right] \\\\P+Q=A

Hence A is written as sum of symmetric matrix (P) and skew-symmetric matrix(Q).

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Safeer PP

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