exactly there is relation. 

2. What are the possible real eigenvalues of a 4 by 4 permutation matrix? (Hint: consider

such a matrix P and powers I, P, P2

, P3

, . . .. Show it eventually has to repeat). You

do not need to show the values you give are all possible, but you still must get the

correct range to achieve full credit.

Solution. Consider where P sends v = (x, y, z, w)

T

. There are only 24 possibilities

so the list v, P v, . . . , P24v must repeat at some point. If v were an eigenvector with

eigenvalue λ, this means v, λv, λ2v, . . . must repeat at some point and λ

k = λ

k+l

for

some k and l. Since P is invertible, λ 6= 0 and we can divide to get λ

l = 1 for some

l. For λ to be real, it must be 1 or −1.

Alternative solution. Or, note that an n × n permutation matrix P is orthogonal,

P

TP = I. For any orthogonal matrix Q, assume Qv = λv. Then also v

TQT = λvT

,

and right-multiplying each side by Qv and λv, respectively, and using QTQ = I, we

get v

T v = v

TQTQv = λ

2v

T v. But v

T v 6= 0 when v is an eigenvector and thus λ

2 = 1,

so λ ∈ {±1}.

A permutation matrix is any $n \times n$ matrix that has exactly one 1 in each row and column, with all other entries being 0. Here is an example of a $6 \times 6$ permutation matrix:

\begin{displaymath}P = \pmatrix{0 &0 &1 &0 &0 &0 \cr

0 &0 &0 &0 &0 &1 \cr

0 &0...

...&0 &0 &0 &0 &0 \cr

0 &0 &0 &0 &1 &0 \cr

0 &1 &0 &0 &0 &0 \cr}\end{displaymath}

All the eigenvalues of a permutation matrix lie on the (complex) unit circle, and one might wonder how these eigenvalues are distributed when permutation matrices are chosen at random (that is, uniformly from the set of all $n \times n$ permutation matrices). Some work has already been done in studying the eigenvalues of permutation matrices. Diaconis and Shahshahani [3] looked at the trace (sum of the eigenvalues), and Wieand [5],[4] investigated the number of eigenvalues that lie in a fixed arc of the unit circle. In both cases, the asymptotic behavior for large n was determined.

Roughly speaking, the number of eigenvalues that lie in a fixed interval on the unit circle will be proportional to the size of the interval and to the dimension n of the matrix. In this paper, the goal will be to allow n to increase while decreasing the size of the interval, so that the number of eigenvalues lying in it should remain fairly constant on average. In particular, we look at the number of eigenvalues Xn,a lying in the interval $I_n = \left(e^{2 \pi i a},\,e^{2 \pi i(a + l/n)}\right]$

actually it is related whether the permutation matrix is circulant.

Dear Prof. Cenap,

Thank you for useful comments, I would like to know, for any square matrix which

Ax = λx, where x ≠ 0.

What does the following result mean?

PAx  = λy, where P is a permutation matrix.

The matrix is real or complex

Up to you consider. I happy to know the results.

Exactly the permutation matrix P is a circulant matrix, as Prof. Cenap commented above.

Suppose that

Ax = λx, where x ≠ 0.

Consider  

PAPTPx = λPx, 

we write

By = (PAP-1)Px = λy, that is  y = Px ≠ 0.

or

By = λy,

the matrix B is similar (congruent) to A via P, so y is an eigenvector of B corresponding  to λ,  any thing else?

Dear Prof. Peter,

Thank you for useful comments. 

Given a square matrix A, which I known the characteristic polynomial, minimal polynomial, eigenvector formula  of any eigenvalue of A, and the inverse formula of A, etc. If we transforms  A to a new matrix B with premultiply A by a permutation matrix P. The new matrix B := PA  has a good forms such as a doubly companion matrix,  Leslie matrix, and so on. How can we use some properties of A for solve the same problems in B?

Suppose

Ax = λx, where x ≠ 0.

I would like to have the meaning of

PAx  = λy,   y = Px,  where  P is a permutation matrix.

If you have any information related about this problem, please let me know, Sir.

Dear Prof. Peter,

I think Peter,  your posed is very badly posed, because you never see the Heading of my post. What is the aim of my mains problem? you consider only the detail in my sub-posed for make you answer !!

Peter said that "The detail must be irrelevant to your question, whatever it is (it is undefined, because you have not defined your term "flip-eigenvector"), because it relates to the standard (simple) eigenvalue problem."

You never see any information before you post, I defined the term "flip-eigenvector" appeared in my problem.

Let P be a permutation matrix, if we premultiplying above equation by P,

PAx = λPx =: λy, say,

effect flip arrays (reverses the order of the elements) in the vertical direction of vector x.

Please stop Peter, you don't teach English to me.

Dear Sir,

I do not have the linguistic literature. I just want to convey the meaning of mathematical problems and need answers only.

If you can not answer the question Or do not understand the question I was not required to post a comment in any way on.

I wrote this, you clearly do not understand.

The problem is clear, unless you try to pretend not to understand it. If you do not understand, no need to answer it.

I think, we have no relation about the flip-eigenvector (namely y)  y and A in eigenvalue problem. The vector  y  is an eigenvector of PA corresponding to the λ only.

Dear Peter,

A post I though it would not be the core grammar. But you understand, right? I'd just that. Thank you for teaching me to.

Dear Prof. Peter,

Thank you for useful comments.