# Sumation of symmetric and skew symmetri metrices

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??

## Answers and Replies

vela
Staff Emeritus
Homework Helper
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;
$$A = \begin{pmatrix} 6 & 1 & 5 \\ 1 & -5 & 4 \\ 5 & 4 & -1 \end{pmatrix}$$
Consider skew symmetric metric as "B"
then;
$$B = \begin{pmatrix} 0 & 1 & 5 \\ -1 & 0 & 4 \\ -5 & -4 & 0 \end{pmatrix}$$

Then sum of matrices A and B is;
$$A+B = \begin{pmatrix} 6 & 2 & 10 \\ 0 & -5 & 8 \\ 0 & 0 & -1 \end{pmatrix}$$
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.

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

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

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

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?

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
Homework Helper
Yes, this this correct.