@MartinSleziak Thank you

I see @MartinSleziak but I think why is it that there are some properties of a square matrix which always exists like eigenvalues of a square matrix exists irrespective of invertibility but inverse of a matrix cannot exists for non-invertible matrices that is there is some restriction? perhaps there will be some more generalized result which can say about the properties of the matrix more generally which may have been proved or may be proved if one finds the theme suitable enough.

I don't really know what to say to your last question, it seems a bit unclear and somewhat broad.

The fact that inverse of a matrix cannot exists for non-invertible matrices is basically the definition of invertible matrix, so that should not be much of a surprise.

its like why eigenvalues exists for any square matrix but inverse of a square matrix does not exist for any square matrix, is there any reason for that?

Of course there is reason for that.

In one case the reasons is that there are counterexamples showing that the claim is not true for non-invertible matrices. In the other case the reason is that there is a proof showing that the claim is true for all matrices.

I am not sure whether you can expect some additional insights to this naive and simple observations.

Yes

At least I will say this.

Hm, Ok thank you!!

Every square matrix has characteristic polynomial. And this polynomial is non-zero. (You do not invertibility here.)

If you are working over $\mathbb C$ (or, more generally, any algebraically closed field), this polynomial has $n$ roots.

So you have $n$ eigenvalues if you count algebraic multiplicity.

This illustrates at least a bit that for existence of eigenvalues, the invertibility does not play any role.

In fact, we know that a matrix is non-invertible iff zero is an eigenvalue.

I see, thank you @MartinSleziak

is there any visualization of Jordan canonical form

like i tried to search jordan forms , applications , intuition, visualization

@BAYMAX Yes, of course.

It depends a bit on what you mean by simultaneously. OR? AND?

AND

Here are all questions that have tags jordan-normal-form and applications: math.stackexchange.com/questions/tagged/…

Here are all questions that have tags jordan-normal-form or applications: math.stackexchange.com/questions/tagged/…

how you typed joran-normal form tag here in chat?

how you got that block?

grey block?

Of course you can do this with any tags, you can use more than two tags. And you can include additional keywords to restrict the search.

A random example: [sequences-and-series] [trigonometry] composition.

BTW there is a separate room for discussing searching and asking about search related topics: In the search of a question.

You can look at the first day of transcript to see what were the intentions when the chatroom was created: chat.stackexchange.com/transcript/46148/2016/10/1

@BAYMAX You can check for yourself by looking at source of the message. Just click on the history in the transcript to get to this: chat.stackexchange.com/messages/44843544/history

In any case, the syntax is [tag:linear-algebra] linear-algebra, [meta-tag:rant] rant.

[tag: linear-algebra]

?

I could not do it?

@BAYMAX It seems that you have left a space between tag: and linear-algebra.

linear-algebra

Wow done

thanks

math.stackexchange.com/questions/tagged/jordanform+intuition

it seems no posts yet

It seems that at the moment there is a single post tagged jordan-normal-form+applications. Only a single post tagged jordan-normal-form+intuition.

on intuitive thinking about Jordan form

So a Jordan Normal form is the same as Jordan canonical form

?

@BAYMAX The tag name is (jordan-normal-form) and not (jordanform).

@BAYMAX As far as I know they are two names for the same thing.

You're too quick for me in switching between various topics.

Sorry

How did you think when you got first introduced to Jordan canonical form?

No post tagged jordan-normal-form+visualization.

@MartinSleziak

@BAYMAX No problem. I have realized long time ago that I have problems in keeping up with younger generation.

@BAYMAX Probably the best answer is confused. I'll leave this question for somebody who knows more about this. If you wish, you can try also the new linear algebra chatroom.

@MartinSleziak I don't think that, a bit time ago when you helped me a lot in functional analysis and measure theory i was not able to keep pace with you :)

@MartinSleziak Ok, thank you for your time

See you later!

see yaa