Dimensions of Matrices Range (equalities).

[GoodSpirit]

Hello everyone,

I’d like to find the following range equalities:
Considering the following:
$$A=B+C \\ A=B.C^T \\ A=[ B^T C^T ]^T$$
I would like to find the function f for each equality above.
$$.\\ dim( R(A) ) = f( R(B) , R(C) )\\$$
Considere that all matrices have compatible dimensions.
R(X) function means range of the column space expressed in the matrix X.
Can you help me?

I sincerely thank you!

All the best

GoodSpirit

 

[conquest]

Hi!

Pardon, could you be a bit more precise about what it is you mean. I have some questions after which I will gladly try and help you out with this problem:

What sort of objects are A, B and C?
Should all the equations hold at the same time or are you asking 3 questions?
What is a range equality, the only range I know of is (in the case of linear algebra) the column space (maybe you mean rank)?
Do you mean by R(X) the column space of X?

 

[GoodSpirit]

Hi Conquest,

A,B and C are rectangular matrices and there are 3 questions. I'm trying to find a function f for each case.
R(X) is the range of the column space defined by the matrix X.
equality is and equation or set of equations that relate variables somehow. Like this

This is the equation that answer the first case
$$\dim(R(A))=\dim(R(B))+\dim(R(C))-\dim(R(B) \cap R(C))$$

This is the equation that answer the third case

$$\dim(R(A))=\dim(R(B))+\dim(R(C))-\dim(R(B^T) \cap R(C^T))$$

The second one is more difficult but is the most important

all the best

GoodSpirit

 

[conquest]

Hey,

I will start working them out right away. The first one you solved there is correct. Since the column space of B+C is just the sum of the column spaces of B and C respectively and the formula you have there is indeed the formula for the dimension of the dimension of a sum of subspaces.

I don't get how you would do the third one though since to me the last part makes no sense. I mean the colum spaces of $B^T$ and $C^T$ need not for general recangular matrices be subspaces of the same space. So how can you take an intersection between them! Do you mean something exotic by the square brackets?

Also the Dim(R(A)) is what you call the rank of a matrix A. It is not hard to prove that in fact the rank of A and $A^T$ is always the same. So for this purpose taking the transpose in the last line is not very relevant.

Anyway I will start to try the problem now. however I am very curious how you would prove the solution you have to the third one since for me it doesn't even make senes.

Best

 

[conquest]

Hello again!

All right I can get back to you quite quickly since I have the idea that there is not in general the formula you are looking for. Since it is quite easy to find for a given matrix B in the second and third cases a two different matrices C and C' such that the rank of both of these is the same but the rank of the corresponding A is not the same (just try it).

A formula about the rank of the matrix A would try to express it in terms of the ranks of the other two matrices and possibly the dimension of the domain or codomain. But apparently if all of these are equal for two cases the rank of A can still be different. So that's why I think there is no general formula (although I have not proved this).

However all is not lost since the first one is still quite correct and we have some inequalities.
For instance we allways have

$Dim(R(A))=Dim(R(A^T))$

and

$Dim(R(AB))\leq min(Dim(R(A)),Dim(R(B)))$

and also (a bit trickier to prove then the last ones but still not too hard)

$Dim(R(AB))\geq Dim(R(A))+Dim(R(B))-n$

for A a m x n matrix and B a n x k matrix

So this yields for the second formula that for B an m x n matrix and C an k x n matrix

$Dim(R(B))+Dim(R(C))-n\leq Dim(R(A)) \leq min(Dim(R(B)),Dim(R(C)))$




and for the third formula we find with B an n x m matrix and C an k x n matrix

$Dim(R(B))+Dim(R(C))-n\leq Dim(R(A)) \leq min(Dim(R(B)),Dim(R(C)))$


may I suggest by the way http://en.wikibooks.org/wiki/Linear_Algebra/Matrix_Multiplication/Solutions problem 15 in particular.

There are actually a lot more very useful equalities and inequalities about the rank of a matrix so you should probably just search for rank of a matrix and mulitplication to get some references like this one http://www.m-hikari.com/ija/ija-password-2007/ija-password9-12-2007/dongIJA9-12-2007.pdf
But I suspect that usually for an identity you would need some extra information in the cases two and three.

I hope this helps.

Cheers

 

[GoodSpirit]

Hi there,

Thank you for answering.

Actually I found the answers in this book

"Matrix Analysis and Applied Linear Algebra"

I´ve been busy lately in a paper writing sorry for not answer sooner.

All the best

GoodSpirit.