I A regular matrix <=> mA isomorphism

Math_QED

Science Advisor
Homework Helper
1,158
383
Hello all

Let ##m_A: \mathbb{K^n} \rightarrow \mathbb{K^n}: X \mapsto AX## and ##A \in M_{m,n}(\mathbb{K})##
(I already proved that this function is linear)

I want to prove that:

A regular matrix ##\iff m_A## is an isomorphism.

So, here is my approach. Can someone verify whether this is correct?

##\Rightarrow##

##A## is a regular matrix, thus ##A^{-1}## exists
Then is ##m_A## an isomorphism, because we can find an inverse function ##m_A^{-1}: X \mapsto A^{-1}X## It's straightforward to see that this function is the inverse. (##m_A \circ m_A^{-1} = 1_{\mathbb{K^n}} ## and ##m_A^{-1} \circ m_A = 1_{\mathbb{K^n}}##)

##\Leftarrow##

##m_A## is an isomorphism, thus ##m_A## has an inverse ##m_A^{-1}: X \mapsto BX##.

When we apply the definition of inverse function, we would deduce that:

##AB = I_n## and ##BA = I_n##. Then it follows, by definition of the inverse matrix, that ##B = A^{-1}##. Thus ##A^{-1}## must exist, because the inverse is unique and there is only one possibility to construct this inverse. Therefore, A is a regular matrix.

Thanks in advance.
 

fresh_42

Mentor
Insights Author
2018 Award
11,153
7,655
Hello all

Let ##m_A: \mathbb{K^n} \rightarrow \mathbb{K^n}: X \mapsto AX## and ##A \in M_{m,n}(\mathbb{K})##
(I already proved that this function is linear)

I want to prove that:

A regular matrix ##\iff m_A## is an isomorphism.

So, here is my approach. Can someone verify whether this is correct?

##\Rightarrow##

##A## is a regular matrix, thus ##A^{-1}## exists
Then is ##m_A## an isomorphism, because we can find an inverse function ##m_A^{-1}: X \mapsto A^{-1}X## It's straightforward to see that this function is the inverse. (##m_A \circ m_A^{-1} = 1_{\mathbb{K^n}} ## and ##m_A^{-1} \circ m_A = 1_{\mathbb{K^n}}##)
Correct. Only "##A## regular ##\Longrightarrow \, n=m##" could be added.
##\Leftarrow##

##m_A## is an isomorphism, thus ##m_A## has an inverse ##m_A^{-1}: X \mapsto BX##.
Why can be assume, that ##m_A^{-1}## is of the form ##m_B##? And again, why is ##n=m##?
When we apply the definition of inverse function, we would deduce that:

##AB = I_n## and ##BA = I_n##. Then it follows, by definition of the inverse matrix, that ##B = A^{-1}##. Thus ##A^{-1}## must exist, because the inverse is unique and there is only one possibility to construct this inverse. Therefore, A is a regular matrix.

Thanks in advance.
 

Math_QED

Science Advisor
Homework Helper
1,158
383
Correct. Only "##A## regular ##\Longrightarrow \, n=m##" could be added.

Why can be assume, that ##m_A^{-1}## is of the form ##m_B##? And again, why is ##n=m##?
n = m because ##AX \in \mathbb{K^n}##, by definition of ##m_A##.

Your second question made me realise that I cannot assume that. I would need to do something like:

##(m_A \circ m_A^{-1})(X) = X \iff m_A(m_A^{-1}(X)) = X \iff Am_A^{-1}(X) = X \Rightarrow m_A^{-1}(X) = BX ## such that## AB = I_n##
And analogue for ##(m_A^{-1} \circ m_A)(X)##?
 

fresh_42

Mentor
Insights Author
2018 Award
11,153
7,655
The basic difficulty here (at least mine) is to keep the parts apart. Since it is almost obviously true, I find it difficult, to separate both sides.
##n = m## because ##AX \in \mathbb{K^n}##, by definition of ##m_A##.
Isn't this the same as to start with ##A \in \mathbb{M}_{n,n}(\mathbb{K})## in the first place? Only demanding ##n=m##?
I would have expected two arguments, that regular matrices as well as isomorphisms can only be established between vector spaces of the same dimension. Of course this is true (in the finite dimensional case), but is it already known or part of the proof? I would have started with "Let ##m_A : \mathbb{K}^n
\rightarrow \mathbb{K}^m## given by ##X \mapsto AX## ..." because that is how you defined ##A##! In this case ##n=m## would have to be shown in both directions of the statement.

##Am^{-1}_A(X) = X \Rightarrow m^{-1}_A (X) = BX## such that ##AB=I_n##
Yes, this is the crucial part. That there cannot be two different inverse elements from the right and from the left should be shown separately in an auxiliary proposition because it is needed everywhere (IMO).

To do this I would consider ##(Am^{-1}_A - I_n)(X)## and apply the definitions of ##0_n## and next that of inverse elements.
 

Math_QED

Science Advisor
Homework Helper
1,158
383
The basic difficulty here (at least mine) is to keep the parts apart. Since it is almost obviously true, I find it difficult, to separate both sides.

Isn't this the same as to start with ##A \in \mathbb{M}_{n,n}(\mathbb{K})## in the first place? Only demanding ##n=m##?
I would have expected two arguments, that regular matrices as well as isomorphisms can only be established between vector spaces of the same dimension. Of course this is true (in the finite dimensional case), but is it already known or part of the proof? I would have started with "Let ##m_A : \mathbb{K^n}
\rightarrow \mathbb{K}^m## given by ##X \mapsto AX## ..." because that is how you defined ##A##! In this case ##n=m## would have to be shown in both directions of the statement.


Yes, this is the crucial part. That there cannot be two different inverse elements from the right and from the left should be shown separately in an auxiliary proposition because it is needed everywhere (IMO).

To do this I would consider ##(Am^{-1}_A - I_n)(X)## and apply the definitions of ##0_n## and next that of inverse elements.
Hm, I have been thinking about this and I believe I might have found an easier explanation

The ##\Rightarrow## was clear. For the ##\Leftarrow## part, we know that a matrix is regular if and only of it's the matrix of an isomorphism (after we choose certain basises), and this is equivalent with saying that ##A^{-1}## exists. Well, now consider ##[m_A]_{E,E}##, with E standard basis of ##\mathbb{K^n}##. Then we see that ##A = [m_A]_{E,E}##. Thus, A is regular and this is what we wanted to show.

Now, we can verify that ##m_A^{-1}: X \mapsto A^{-1}X## is the inverse, by applying the definition and regarding the fact that ##A^{-1}## exists and the inverse function is unique.

What do you think about this?
 
Last edited:

fresh_42

Mentor
Insights Author
2018 Award
11,153
7,655
I still struggle to keep assumption and implication apart. So let me proceed in small steps.
For the ##\Leftarrow## part,...
This means: Given an isomorphism ##m_A## defined by ##m_A(X)=AX##.

So if we denote the matrix of ##m_A## according to the basis ##E=\{E_1,\ldots,E_N\}## by ##[m_A]##, we have ##[m_A](E_i)=AE_i## by definition of ##m_A## and thus ##[m_A]=A##.
...we know that a matrix is regular if and only of it's the matrix of an isomorphism (after we choose certain basises), and this is equivalent with saying that ##A^{-1}## exists.
Now, why isn't this exactly what we have to show: ##m_A## isomorphic ##\Longrightarrow A = [m_A]## regular? Somewhere the existence of ##m_A^{-1}## has to be used and we cannot assume, yet, that ##[m_A^{-1}]=A^{-1}## or that ##A^{-1}## even exists. Only ##[m_A^{-1}]##.

(This were my thoughts, and I hope I didn't confuse something, because the longer I think about it, the more I get confused ... The entire exercise sounds like the question about hen and egg.)
 

Math_QED

Science Advisor
Homework Helper
1,158
383
Well, let me write out everything more clearly.

Definitions:

1) An ##n## x ##n## matrix ##A## is regular if it's the matrix of an isomorphism, or equivalent: if ##A^{-1}## exists.
2) ##m_A: \mathbb{K^n} \rightarrow \mathbb{K^n}: X \mapsto AX##

Theorem: A regular matrix ##\iff m_A## is an isomorphism.

Proof
:

##\Rightarrow## If ##A## is a regular matrix, then ##m_A## is an isomorphism.

If A is regular, then we know, by definition, that ##A^{-1}## exists. I now claim that ##m_A^{-1}: X \mapsto A^{-1}X##

This claim is true, because ##m_A \circ m_A^{-1} = 1_{\mathbb{K^n}} ## and ##m_A^{-1} \circ m_A = 1_{\mathbb{K^n}}##
and thus this function is the (unique) inverse. Because ##m_A## has an inverse, we know that ##m_A## is bijective and thus an isomorphism (It's clear that ##m_A## is a linear function)

##\Leftarrow## If ##m_A## is an isomorphism, then ##A## is a regular matrix.

##m_A## is an isomorphism, thus every matrix associated with it is regular. ##[m_A]_{E,E} = A## when we choose that ##E## is the standard basis of ##\mathbb{K^n}##. Thus, because ##A## is the matrix of an isomorphism, ##A## must be regular (by definition) and this ends the proof.

QED.
 

fresh_42

Mentor
Insights Author
2018 Award
11,153
7,655
##m_A## is an isomorphism, thus every matrix associated with it is regular.
Why? And why isn't this exactly what has to be shown? ##m_A## is an isomorphism only guarantees another isomorphism ##\varphi## such that ##\varphi m_A = 1##. If you already know, that these belong to regular matrices, why to prove something at all?
 

Math_QED

Science Advisor
Homework Helper
1,158
383
Why? And why isn't this exactly what has to be shown? ##m_A## is an isomorphism only guarantees another isomorphism ##\varphi## such that ##\varphi m_A = 1##. If you already know, that these belong to regular matrices, why to prove something at all?
By definition! And you don't know that A is a matrix associated with ##m_A## until you have found a basis ##B## such that ##[m_A]_{B,B}## = A. But such a basis exists, as it is the standard basis.
 

fresh_42

Mentor
Insights Author
2018 Award
11,153
7,655
By definition! And you don't know that A is a matrix associated with ##m_A## until you have found a basis ##B## such that ##[m_A]_{B,B}## = A. But such a basis exists, as it is the standard basis.
We know by construction of ##m_A## as ##m_A(X)=AX## that ##[m_A]_{B,B} = A## by applying unit vectors. But how do you get from ##\varphi m_A = 1## to ##\varphi(X) = A^{-1}X##? In this direction we only have ##\varphi## and know nothing about a matrix.
##m_A## is an isomorphism, thus every matrix associated with it is regular.
(why?) and
Thus, because ##A## is the matrix of an isomorphism, ##A## must be regular (by definition).
is what I don't see. By definition we only get ##\varphi##, no matrix of ##\varphi##. If we automatically have matrices, then there is nothing left to show, because it's clear then that:
$$m_A \text{ isomorphism with matrix } A \Leftrightarrow m_A^{-1} \text{ isomorphism with matrix } B \Leftrightarrow AB=1 $$
To me the bridge between regular matrices and isomorphisms is what it's all about, i.e. why is ##[\varphi] = A^{-1}##.
 

Math_QED

Science Advisor
Homework Helper
1,158
383
We know by construction of ##m_A## as ##m_A(X)=AX## that ##[m_A]_{B,B} = A## by applying unit vectors. But how do you get from ##\varphi m_A = 1## to ##\varphi(X) = A^{-1}X##? In this direction we only have ##\varphi## and know nothing about a matrix.

(why?) and

is what I don't see. By definition we only get ##\varphi##, no matrix of ##\varphi##. If we automatically have matrices, then there is nothing left to show, because it's clear then that:
$$m_A \text{ isomorphism with matrix } A \Leftrightarrow m_A^{-1} \text{ isomorphism with matrix } B \Leftrightarrow AB=1 $$
To me the bridge between regular matrices and isomorphisms is what it's all about, i.e. why is ##[\varphi] = A^{-1}##.
In the ##\Leftarrow## part I never use the existence of ##m_A^{-1}## (check my new proof that I wrote 2 posts ago). We know from the definition that every matrix of an isomorphism is regular. That's the only thing we use. We choose the standard basis to show that the isomorphism ##m_A## can be represented by ##A## and we know from the definition that, when this is possible, A is regular. And, yes we have shown it's possible.
 

fresh_42

Mentor
Insights Author
2018 Award
11,153
7,655
In the ##\Leftarrow## part I never use the existence of ##m_A^{-1}## (check my new proof that I wrote 2 posts ago).
But that's all you have. How can you not use it?
We know from the definition that every matrix of an isomorphism is regular.
In this case I fold. If you have such a definition, then all of the above is trivial since ##m_A(X)=AX## implies that there is already a basis (to express ##A##), and all which is left to show is, that there cannot be two different matrices.
 

Want to reply to this thread?

"A regular matrix <=> mA isomorphism" You must log in or register to reply here.

Related Threads for: A regular matrix <=> mA isomorphism

  • Posted
Replies
5
Views
3K
  • Posted
Replies
16
Views
9K
Replies
5
Views
30K
  • Posted
Replies
5
Views
1K
  • Posted
Replies
5
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top