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#1 2020-05-13 23:47:43

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Matrices

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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#2 2020-05-14 00:31:52

Bob
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Registered: 2010-06-20
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Re: Matrices

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 smile

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#3 2020-05-22 12:54:17

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

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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#4 2020-05-22 20:52:53

Bob
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Re: Matrices

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 smile

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#5 2020-05-25 23:23:38

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

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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#6 2020-05-26 00:14:02

Bob
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Re: Matrices

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 smile

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#7 2020-05-26 22:06:48

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

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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#8 2020-05-26 23:13:11

Bob
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Posts: 8,908

Re: Matrices

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


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 smile

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#9 2020-06-05 16:58:47

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

Would you please explain the following topics:
(1) elementary operations(transformations) of a matrix
(2) invertible matrices, inverse of a matrix by elementary operations


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

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#10 2020-06-05 21:25:05

ganesh
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Re: Matrices


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#11 2020-06-06 18:17:22

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

how to find inverse of matrix  using elementary operations of an matrix and without using determinants?


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

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#12 2020-06-06 19:25:34

Bob
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Re: Matrices

The link you were given before does this:

Inverse of a Matrix using Elementary Row Operations.

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 smile

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#13 2020-06-08 02:02:24

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

Why does the method of finding inverse operations using elementary operations looks so hard and what should I do to reduce the mistakes while doing this method? Especially for  3x3 matrices?

Last edited by 666 bro (2020-06-08 02:03:00)


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

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#14 2020-06-08 19:35:32

Bob
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Posts: 8,908

Re: Matrices

With any area of maths that is causing me difficulties I take a step back to easier questions first.  How are you getting on with applying the method for 2 x 2 matrices?  If that's OK, try 3 x 3 with simple numbers in the cells … maybe have some lines with just a single 1 and the rest zeros.  Build up to more complicated cases gradually.

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 smile

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#15 2020-06-12 16:09:57

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

This is what I know about a inverse of matrices using row operations. please ask me questions to challenge my understanding?


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

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#16 2020-06-12 21:16:33

Bob
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Registered: 2010-06-20
Posts: 8,908

Re: Matrices

hi 666bro

Ok.  Here's Q1:

Use elementary operations to find the inverse of

Check by multiplying that your answer is correct.  To post the answer use square brackets math and /math commands with the matrix like this:

\begin{pmatrix}
a & b\\
b & c
\end{pmatrix}


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 smile

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#17 2020-06-24 03:13:49

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,956
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Re: Matrices

Just a thought: People should really learn linear algebra, not "matrices"


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#18 2020-06-24 04:31:02

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

But I'm a 12th class student. How would I start learning linear algebra at a time?


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

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#19 2020-06-25 05:14:21

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Re: Matrices

What does the phrase "at a time" mean?

I had been a 12th class student about 4-5 years ago, so I feel you. A good resource for learning Linear Algebra is "Linear Algebra Done Right". Also, Alon Amit's Abstract Motivated Linear Algebra seems very approachable, except that not many chapters have been published.

Linear algebra is the study of a kind of mathematical object called "vectors" and also nice transformations of vectors (e.g, rotation, scaling, reflection). Matrices are a specific mechanism that lets one describe and manipulate these transformations. As I understand, the way matrices are taught in the 12th standard is ill-motivated. They project matrices as the central object of study, rather than vectors. As a result, they end up teaching students a very mechanical view of the subject -- the students tend to think that they need to learn to compute determinants, transposes, inverse of matrices, without understanding what any of that is good for.

As far as I can tell, all subfields of STEM use linear algebra in some form or the other. That includes pure mathematicians, computer scientists, engineers, physicists. Having a good understanding of linear algebra is of immense value.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#20 2020-06-25 13:00:27

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

Ok, I want to learn linear algebra as a beginner so, where should I start learning?


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

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#21 2020-06-25 16:55:38

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,956
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Re: Matrices

Agnishom wrote:

A good resource for learning Linear Algebra is "Linear Algebra Done Right". Also, Alon Amit's Abstract Motivated Linear Algebra seems very approachable, except that not many chapters have been published.

You can also look into https://www.codingthematrix.com/ if that sort of thing interests you


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#22 2020-06-25 16:56:38

666 bro
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From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

Is it for beginners? and I'm not familiar with coding.

Last edited by 666 bro (2020-06-25 17:01:14)


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

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#23 2020-06-26 02:40:22

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,956
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Re: Matrices

The books I mentioned are for beginners, yes. Not sure about 'coding the matrix' but it might have an intro to coding as well


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#24 2020-06-26 03:03:23

666 bro
Member
From: Flatland
Registered: 2019-04-26
Posts: 508

Re: Matrices

Will miscellaneous exercises of matrices in ncert maths textbook helps us for understanding concrete concepts?


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

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#25 2020-06-26 14:16:19

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,956
Website

Re: Matrices

What are concrete concepts?

My experience is that the Miscellaneous exercises in the NCERT books are pretty good for testing your problem solving abilities. However, the way they are taught in schools often does not encourage students to reflect on what really is happening.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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