Sumation of symmetric and skew symmetri metrices

[harshakantha]

Express \left(\begin{array}{cccc}
6 & 1 & 5\\
-2 & -5 & 4\\
-3 & 3 & -1\
end{array}
\right) as the sum of the symmetric and skew symmetric matrices.

I did this following way

Consider symmetric metric as "A"
then;
A = \left(\begin{array}{cccc}
6 & 1 & 5\\
1 & -5 & 4\\
5 & 4 & -1\
\end{array}
\right)

Consider skew symmetric metric as "B"
then;
B = \left(\begin{array}{cccc}
0 & 1 & 5\\
-1 & 0 & 4\\
-5 & -4 & 0\
\end{array}
\right)

Then sum of matrices A and B is;
A+B= \left(\begin{array}{cccc}
6 & 2 & 10\\
0 & -5 & 8\\
0 & 0 & -1\
\end{array}
\right)

is this correct??

 

[vela]

Fixing your LaTeX...

Express
\begin{pmatrix}
6 & 1 & 5 \\
-2 & -5 & 4 \\
-3 & 3 & -1
\end{pmatrix}
as the sum of the symmetric and skew symmetric matrices.

I did this following way

Consider symmetric metric as "A"
then;
A = \begin{pmatrix}<br /> 6 &amp; 1 &amp; 5 \\ <br /> 1 &amp; -5 &amp; 4 \\ <br /> 5 &amp; 4 &amp; -1 <br /> \end{pmatrix} <br />
Consider skew symmetric metric as "B"
then;
<br /> B = \begin{pmatrix}<br /> 0 &amp; 1 &amp; 5 \\ <br /> -1 &amp; 0 &amp; 4 \\ <br /> -5 &amp; -4 &amp; 0<br /> \end{pmatrix} <br />

Then sum of matrices A and B is;
A+B = \begin{pmatrix}<br /> 6 &amp; 2 &amp; 10 \\ <br /> 0 &amp; -5 &amp; 8 \\ <br /> 0 &amp; 0 &amp; -1<br /> \end{pmatrix}<br />
is this correct??

No. The problem is asking you to find A and B such that A+B is equal to the original matrix. This is obviously not the case for your A and B.

 

[harshakantha]

Thank you for fixing Latex vela , oh.. I think I've understood the question wrongly, so can you give me a hint, on how to do that in correct way

 

[I like Serena]

Let's take a generic symmetric and a skewed symmetric matrix.

Say:
A=\begin{pmatrix}a &amp; b \\ b &amp; d \end{pmatrix}\qquad A=\begin{pmatrix}0 &amp; q \\ -q &amp; 0\end{pmatrix}

Adding them up will yield:
A+B=\begin{pmatrix}a &amp; b+q \\ b-q &amp; d \end{pmatrix}

You should note that the average of (b-q) and (b+q) is b.

Now can you think up how to construct a symmetric and a skewed symmetric matrix from a given matrix?

 

[harshakantha]

Thank you I like Serena, I've found a formula to express a square matrix by using symmetric and skew symmetric matrices here it is;

Let A be the given square matrix
A can be uniquely expressed as sum of a symmetric matrix and a skew symmetric matrix, which is

A =(A+A')/2 + (A-A')/2 consider A' is Transpose of matrix A;
by using this I was able to got the symmetric matrix and a skew symmetric matrix for the given matrix.. is this correct?

 

[I like Serena]

Yes, this this correct.