Linear Algebra Matrix Proof problem (1 Viewer)

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

10
0
1. The problem statement, all variables and given/known data
Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=Im


3. The attempt at a solution
I'm assuming I would need to start with the def. That there exists P an mxm invertible matrix and Q an nxn invertible matrix s.t. A=P(Im 0)Q

then P(Im 0)Q B = Im

now I might multiply left hand side by P inverse and right hand side by Q inverse.

I'm stuck am I going in the right direction???
 

Office_Shredder

Staff Emeritus
Science Advisor
Gold Member
3,718
98
I'm a bit confused by the notation here. What is Im 0? I'm assuming Im is the mxm identity matrix
 
10
0
Yes Im is the mxm identity matrix and 0 is the mxn zero matrix.

I'm thinking of going another direction.

What if I start with letting B be any nxm matrix with rank n. Then AB would be an mxm matrix with rank m, then by the Thm (in my book) 2.18 corollary 2 an nxn matrix is invertible iff its rank is n. AB is invertible. Let M be the inverse then (AB)M=Im which means
A(BM)=Im therefore BM would be the nxm matrix we are looking for.

What do you think? Would that do it, any holes?
 
31,919
3,891
Yes Im is the mxm identity matrix and 0 is the mxn zero matrix.
As Office_Shredder asked, what does Im 0 mean? Inquiring minds want to know.
 
10
0
Im (I subscript m) is the mxm identity matrix and 0 is the m x (n-m) zero matrix.
 
31,919
3,891
Then Im 0 is the product of Im and the m x (n - m) zero matrix, which is 0. Were you thinking that Im times a zero matrix is something other than the same zero matrix?
 
10
0
No. How about this? A= (Im 0)P where P is an nxm invertible matrix. Then replace A with (Im 0)P(B) = Im then (Im 0) P P^-1(Im)=Im
(0 ) (that is a column block matrix with Im being the Identity matrix and 0 being a zero matrix (n-m) x m) then that would make
B = p^-1 (Im)
(0 ) That is a partitioned matrix Im is mxm and the zero is (n-m) x m
Will that work?
 

Office_Shredder

Staff Emeritus
Science Advisor
Gold Member
3,718
98
(that is a column block matrix with Im being the Identity matrix and 0 being a zero matrix (n-m) x m
Is this what Im 0 is? It's hard to tell
 
31,919
3,891
I think I have figured out what you're trying to communicate, but your notation was no help. What you are writing as (Im 0) looks to me like a matrix product, and what you meant was the m x n matrix (Im|0).
 

The Physics Forums Way

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