Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2012-11-19 21:00:09

princess snowwhite
Member
Registered: 2012-11-06
Posts: 29

matrix

let P be any n*n square matrix whose row sum equals 1 then for any postive m the row sum of the matrix P^m equals 1 ,state true or false

Offline

#2 2012-11-19 21:57:13

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: matrix

Hi princess snowwhite;

I am not sure about the term row sum. If you mean that each row sums to 1 then the matrix is called stochastic and is used in Markov processes. Every integer power of that matrix will also be stochastic meaning the rows sum to 1. So, I would say yes.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

#3 2012-11-19 21:58:48

princess snowwhite
Member
Registered: 2012-11-06
Posts: 29

Re: matrix

Yes It Does Mean That Each Roq Sums To 1

Offline

#4 2012-11-19 22:03:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: matrix

Then I would say the answer is yes but I do not remember how to prove that.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

#5 2012-11-19 22:09:00

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: matrix

hi princess snowwhite

Just proved it is true for all 2 x 2 matrices (algebraically).  Don't think I can generalise that method of proof though.  sad

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

Offline

#6 2012-11-19 22:17:32

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: matrix

Just gone a step further, which may then lead to an inductive proof.

If A and B are 2 x 2 matrices having the property, the AB has the property.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

Offline

#7 2012-11-20 01:59:20

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: matrix

I think I can put together a proof for n x n if you wish.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

Offline

#8 2012-11-20 02:02:49

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: matrix

Hi Bob

How would you do an iinductive proof here?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#9 2012-11-20 03:44:26

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: matrix

Good question.  I abandoned that idea and went for a aij type proof instead.  It's just in my head so who knows how it will look when it goes down in type.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

Offline

Board footer

Powered by FluxBB