Sum of symmetric and skew symmetric matrix problem

diogomgf

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Not sure how to answer the following problem:

Show that any square matrix can be written as the sum of a symmetric and a skew symmetric matrix.
 
Not sure how to answer the following problem:

Show that any square matrix can be written as the sum of a symmetric and a skew symmetric matrix.
Please follow the rules of posting in this forum - enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/

Please share your work/thoughts with us - so that we know where to begin to help you. Also include the context - topic being taught in class now.

Please post the EXACT problem as it was presented to you.

To start with, please answer:

What is symmetric matrix?

What is skew-symmetric matrix?
 
Please follow the rules of posting in this forum - enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/

Please share your work/thoughts with us - so that we know where to begin to help you. Also include the context - topic being taught in class now.

Please post the EXACT problem as it was presented to you.

To start with, please answer:

What is symmetric matrix?

What is skew-symmetric matrix?

A symmetric matrix is a matrix such that the transposed of that matrix is the matrix itself.
The skew symmetric matrix is a matrix such that the transposed of that matrix is the negative matrix itself...

The problem is presented as in the OP.
 
Not sure how to answer the following problem:
Show that any square matrix can be written as the sum of a symmetric and a skew symmetric matrix.
HINTS: Suppose that \(\displaystyle A\) is an \(\displaystyle n\times n\) real matrix.
Consider the matrices \(\displaystyle B=\tfrac{1}{2}(A+A')~\&~C=\tfrac{1}{2}(A-A')\)
 
HINTS: Suppose that \(\displaystyle A\) is an \(\displaystyle n\times n\) real matrix.
Consider the matrices \(\displaystyle B=\tfrac{1}{2}(A+A')~\&~C=\tfrac{1}{2}(A-A')\)

Sorry for only replying a week later.
\(\displaystyle A' \) is the skew symmetric matrix?
 
Sorry that is for transpose. See here.

Thought \(\displaystyle A^T \) was the transpose...

I've been slowly studying linear algebra on my spare time, following a very dense book and I haven't reach vector spaces yet... So I've no idea where you are trying to go with that example you posted in the link.


HINTS: Suppose that \(\displaystyle A\) is an \(\displaystyle n\times n\) real matrix.
Consider the matrices \(\displaystyle B=\tfrac{1}{2}(A+A')~\&~C=\tfrac{1}{2}(A-A')\)

\(\displaystyle
B^T = \tfrac{1}{2} (A + A^T) = B
\) .

\(\displaystyle
C^T = - \tfrac{1}{2} (-A + A^T) = -C
\) .

\(\displaystyle B\) is symmetric, \(\displaystyle C\) is skew symmetric. \(\displaystyle B+C = A \) , with \(\displaystyle A\) being a real matrix.
 
Last edited:
Thought \(\displaystyle A^T \) was the transpose... I've been slowly studying linear algebra on my spare time, following a very dense book and I haven't reach vector spaces yet... So I've no idea where you are trying to go with that example you posted in the link.
As you do your study, you will find that two different textbooks can use a symbol in exactly opposite & different ways. That is just the curse of different authors/professors. In grad school at the same university I had three major professors that each used a different notation for set intersection: \(\displaystyle AB,~A\cdot B,\text{ or }A\cap B\).
As for this case it you question does not depend on vector spaces. Use the link to construct symmetric and skew symmetric matrices.
 
As you do your study, you will find that two different textbooks can use a symbol in exactly opposite & different ways. That is just the curse of different authors/professors. In grad school at the same university I had three major professors that each used a different notation for set intersection: \(\displaystyle AB,~A\cdot B,\text{ or }A\cap B\).
As for this case it you question does not depend on vector spaces. Use the link to construct symmetric and skew symmetric matrices.

I'm sorry I had to edit my reply, but I posted by mistake.
I've developed your hint and I think got there?

Cheers,
-Diogo
 
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