Sumation of symmetric and skew symmetri metrices

  • #1
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??:smile:
 

Answers and Replies

  • #2
vela
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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;
[tex]A = \begin{pmatrix}
6 & 1 & 5 \\
1 & -5 & 4 \\
5 & 4 & -1
\end{pmatrix}
[/tex]
Consider skew symmetric metric as "B"
then;
[tex]
B = \begin{pmatrix}
0 & 1 & 5 \\
-1 & 0 & 4 \\
-5 & -4 & 0
\end{pmatrix}
[/tex]

Then sum of matrices A and B is;
[tex]A+B = \begin{pmatrix}
6 & 2 & 10 \\
0 & -5 & 8 \\
0 & 0 & -1
\end{pmatrix}
[/tex]
is this correct??:smile:
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.
 
  • #3
Thank you for fixing Latex vela :smile:, oh.. I think I've understood the question wrongly, so can you give me a hint, on how to do that in correct way :smile:
 
  • #4
I like Serena
Homework Helper
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Let's take a generic symmetric and a skewed symmetric matrix.

Say:
[tex]A=\begin{pmatrix}a & b \\ b & d \end{pmatrix}\qquad A=\begin{pmatrix}0 & q \\ -q & 0\end{pmatrix}[/tex]

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

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?
 
  • #5
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.:smile:. is this correct?
 
  • #6
I like Serena
Homework Helper
6,579
177
Yes, this this correct.
 

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