# On an eigenvector of matrix

• Jul 18th 2012, 07:24 AM
studenttt
On an eigenvector of matrix
A=||A(i,j)|| (i,j=1,…,n) (n>2) is a binary matrix with zero diagonal and A(i,j)=1-A(j,i) for i≠j. W=(1,1,…,1)’ is an eigenvector for matrix B=A*A. Will W be an eigenvector for matrix A too? Why?
• Jul 18th 2012, 09:22 AM
HallsofIvy
Re: On an eigenvector of matrix
Quote:

Originally Posted by studenttt
A=||A(i,j)|| (i,j=1,…,n) is a binary matrix with zero diagonal and A(i,j)=1-A(j,i) for i≠j. W=(1,1,…,1)’ is an eigenvector for matrix B=A*A. Will W be an eigenvector for matrix A too? Why?

Did you try anything at all on this? In particular, what would such a 2 by 2 matrix be like? Is (1, 1)' an eigenvector in that simple case?
• Jul 18th 2012, 12:10 PM
studenttt
Re: On an eigenvector of matrix
Quote:

Originally Posted by HallsofIvy
Did you try anything at all on this? In particular, what would such a 2 by 2 matrix be like? Is (1, 1)' an eigenvector in that simple case?

Just forgot to specify that n>2.
Well, here is example: A=(0,1,0; 0,0,1; 1,0,0) - is a binary matrix 3 by 3, B=A*A=(0,0,1; 1,0,0; 0,1,0) and (1, 1, 1)' is an eigenvector for B and for A too.
• Jul 18th 2012, 01:22 PM
HallsofIvy
Re: On an eigenvector of matrix
But $\begin{bmatrix}0 & 1 & 1 \\ 0 & 0 & 1\\ 0 & 0 & 0\end{bmatrix}\begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}= \begin{bmatrix}2 \\ 1 \\ 0\end{bmatrix}$ does NOT satisfy that even though the matrix does satisfy "A(i,j)=1-A(j,i) for i≠j".
• Jul 18th 2012, 01:44 PM
studenttt
Re: On an eigenvector of matrix
Quote:

Originally Posted by HallsofIvy
But $\begin{bmatrix}0 & 1 & 1 \\ 0 & 0 & 1\\ 0 & 0 & 0\end{bmatrix}\begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}= \begin{bmatrix}2 \\ 1 \\ 0\end{bmatrix}$ does NOT satisfy that even though the matrix does satisfy "A(i,j)=1-A(j,i) for i≠j".

So what? Read the TS question once more - it seems to me you don't understand the question. If $A=\begin{bmatrix}0 & 1 & 1 \\ 0 & 0 & 1\\ 0 & 0 & 0\end{bmatrix}$ it gives nothing for solution because nither A nor B=A*A will have (1,1,1)' as an eigenvector. The problem is: GIVEN that B has (1,1,...,1)' as an eigenvector. FIND (PROOVE OR UNPROOVE): Will A (always) have (1,1,...,1)' as an eigenvector? If yes - it must be proved in some way, if no - we must just find at least one matrix A (binary matrix with zero diagonal and A(i,j)=1-A(j,i) for i≠j) such that W=(1,1,…,1)’ is not its eigenvector while W is an eigenvector for B=A*A.
• Jul 18th 2012, 02:59 PM
HallsofIvy
Re: On an eigenvector of matrix
I did read the question again. You did NOT say "if (1, 1, 1, 1) was an eigenvector of B", you asserted it was. I didn't check that- I assumed you knew what you were saying.
• Jul 19th 2012, 12:51 AM
studenttt
Re: On an eigenvector of matrix
Quote:

Originally Posted by HallsofIvy
I did read the question again. You did NOT say "if (1, 1, 1, 1) was an eigenvector of B", you asserted it was. I didn't check that- I assumed you knew what you were saying.

It was not me who did not say "if" - it is a condition of the problem.
It is just a common form of a problems posting (imho). Suppose the problem of a form "a and b are integer numbers. Will a+b be an integer too?". Will you say "stop.... a=1/2 is not an integer!"?. It is just a condition... every problem has got a condition (words like "if", "suppose", "assume" are meant by default) and a question.
Our condition is "A=||A(i,j)|| (i,j=1,…,n) (n>2) is a binary matrix with zero diagonal and A(i,j)=1-A(j,i) for i≠j. W=(1,1,…,1)’ is an eigenvector for matrix B=A*A."
That is "Suppose we have a binary matrix A=||A(i,j)|| (i,j=1,…,n) (n>2) with zero diagonal and A(i,j)=1-A(j,i) for i≠j. Suppose W=(1,1,…,1)’ is an eigenvector for matrix B=A*A." The question is "Will W (always, under the condition of the problem) be an eigenvector for matrix A too? Why?".
By the way, if you are sure the word "if" is strictly required in the posted problem condition, I'll try to add it there.
• Jul 24th 2012, 02:48 PM
studenttt
Re: On an eigenvector of matrix
Any new ideas?
• Aug 20th 2012, 07:32 AM
studenttt
Re: On an eigenvector of matrix