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**666 bro****Member**- From: Kota
- Registered: 2019-04-26
- Posts: 94

Now I'm learning matrices my question is what does a compact form of an matrix really mean?

"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,829

Wow! Where did that come from?

I have never met this term. I did a bit of internet searching and found two possibilities:

Compact matrix pseudogroups: https://www.researchgate.net/publicatio … eudogroups

and

Compact quantum group: https://en.wikipedia.org/wiki/Compact_quantum_group

It sounds like you'll need to study group theory first.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob Bundy

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**666 bro****Member**- From: Kota
- Registered: 2019-04-26
- Posts: 94

What does the transpose of a matrix mean and how it could be visually seen?

"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,829

hi 666 bro

You turn the rows into columns:

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob Bundy

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**666 bro****Member**- From: Kota
- Registered: 2019-04-26
- Posts: 94

Generally how would we express the given arbitary matrix as the sum of symmetric and skew symmetric matrix?

"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,829

hi 666 bro

I've not met this before so I had to find out what symmetric and skew symmetric mean.

Firstly note: these terms only apply to square matrices: n by n or rows = columns

A matrix is said to be symmetric if it equals its transpose.

and skew symmetric if negating it gives the transpose.

For a symmetric matrix the elements are 'reflected' in the leading diagonal. For a skew, the leading diagonal must have only zeros and the other elements reverse their sign when reflected in the leading diagonal.

example:

and

Suppose B is any square matrix, and consider

The leading diagonal of both B and its transpose are the same, so when subtracted the result has only zeros on that diagonal. The other elements have the property required for a skew matrix, so this new matrix is skew. This is best understood by seeing an example.

Hopefully, you can see how the elements occur again in the reflected positions with their sign changed.

Now consider the matrix 1/2(B + Btranspose). This time the result is a symmetric matrix.

Thus we have a way to make a sum of two matrices,one skew and one symmetric, to make any square matrix.

which is what you are looking for.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob Bundy

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**666 bro****Member**- From: Kota
- Registered: 2019-04-26
- Posts: 94

If A and B are symmetric matrices of same order(nxn) then how could ( AB-BA) be an skew symmetric matrix?, please explain with an example?

"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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**bob bundy****Administrator**- Registered: 2010-06-20
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A 2 x 2 version of a proof is straight forward enough:

Multiplying these swopped around gives

So AB - BA gives

This is skew symmetric.

The general proof for n by n matrices is going to look very much more complicated as it will involve a subscript i,j type elements. Do you want me to have a go?

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob Bundy

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**666 bro****Member**- From: Kota
- Registered: 2019-04-26
- Posts: 94

No, but I understood this 2x2 example and before asking this question I tried to solve and I got mistake in multiplying BA

"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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