Suppose we have $n\times n$ matrix $A$ having only positive elements and satisfying the property $a_{ij}=\dfrac{1}{a_{ji}}$ (a matrix satisfying this property is called a reciprocal matrix).  If its largest eigenvalue $λ_{max}$ is equal to $n$, then the matrix $A$ satisfies the property (consistency property) $a_{ij}a_{jk}=a_{ik}$ where $i,j,k=1,2,...,n$.    I already found an example of $5\times 5$ reciprocal matrix  to show this condition:  \begin{bmatrix}1&amp;1/2&amp;1&amp;1&amp;1/4\\2&amp;1&amp;2&amp;2&amp;1/2\\1&amp;1/2&amp;1&amp;1&amp;1/4\\1&amp;1/2&amp;1&amp;1&amp;1/4\\4&amp;2&amp;4&amp;4&amp;1\end{bmatrix}  It has simple characteristic equation $λ^5-5λ^4=0$, so the eigenvalues of matrix $A$ are $λ=0$ and $λ=5$ and it's shown that $λ_{max}=n=5$.  Is there anyone can help me to give another positive reciprocal matrix example  that has simple characteristic equation and integer number as its eigenvalues ($5\times 5$ or $6\times 6$ or bigger size matrix)?
120572
Suppose we have $n\times n$ matrix $A$ having only positive elements and satisfying the property $a_{ij}=1/a_{ji}$ (a matrix satisfying this property is called a reciprocal matrix).  If its largest eigenvalue $λ_{max}$ is equal to $n$, then the matrix $A$ satisfies the property (consistency property) $a_{ij}a_{jk}=a_{ik}$ where $i,j,k=1,2,...,n$.    I already found an example of $5\times 5$ reciprocal matrix  to show this condition:  \begin{bmatrix}1&amp;1/2&amp;1&amp;1&amp;1/4\\2&amp;1&amp;2&amp;2&amp;1/2\\1&amp;1/2&amp;1&amp;1&amp;1/4\\1&amp;1/2&amp;1&amp;1&amp;1/4\\4&amp;2&amp;4&amp;4&amp;1\end{bmatrix}  It has simple characteristic equation $λ^5-5λ^4=0$, so the eigenvalues of matrix $A$ are $λ=0$ and $λ=5$ and it's shown that $λ_{max}=n=5$.  Is there anyone can help me to give another positive reciprocal matrix example  that has simple characteristic equation and integer number as its eigenvalues ($4\times 4, 5\times 5$ or $6\times 6$ matrix)?
709
Let $M,N$ be $n \times n$ square matrices over an algebraically closed field with the properties that the trace of both matrices coincides along with all powers of the matrix. More specifically, suppose that $\mathrm{Tr}(M^k) = \mathrm{Tr}(N^k)$ for all $1\leq k \leq n$. The following questions about eigenvalues is then natural and I was thinking it would be an application of Cayley-Hamilton but I am having trouble writing out a proof.  How do we show that $M$ and $N$ have the same eigenvalues?  Added (because this question is now target of many duplicates, it should state its hypotheses properly). Assume that all the mentioned values of $k$ are nonzero in the field considered; in other words either the field is of characteristic $0$, or else its prime characteristic $p$ satisfies $p&gt;n$.
86559
I would be very grateful if someone can help me with the following assignment:  I'm given an $n\times n$ matrix $A$.  If $A \neq I,0$ and $A = A^2$, I need to prove that $\lambda=0$ and $\lambda=1$ are $A$'s eigenvalues and that they are $A$'s only eigenvalues.
86102
An $n\times n$ complex matrix $A$ satisfies $A^k=I_n$, where $I_n$ is $n\times n$ identity matrix and $k$ is positive integer $\gt1$. Suppose that 1 is not an eigen value of $A$. Then which of the fallowing is necessarily true?  1) $A$ is diagonalizble  2) $A+A^2+...+A^{k-1}=0$  3) $tr(A)+tr(A^2)+...+tr(A^{k-1})=-n$  4) $A^{-1}+A^{-2}+...+A^{-(k-1)}=-I_n$  I think 1) and 2) are ture.
85937
Prove:     If $v_1,...,v_r$ are eigenvectors that correspond to distinct eigenvalues $\lambda_1, ...,\lambda_r$ of an $n \times n$ matrix $A$, then the set $\{v_1,...,v_r\}$ is linearly independent.   Please give an example and tell me how this theorem works!
21751
If $A$ and $B$ are $3\times 3$ matrices and $A$ is invertible, then can we say that there exists an integer $n$ such that $A+nB$ invertible?  I was trying to show this by choosing $n$ such that eignevalues of $A+nB$ are non-zero. In the case where $B = I$ we can find the eigenvalues of $A+nB$ that would be $\lambda + nB$ (though I am not certain about its proof). This choosing of $n$ such that $\lambda$ is not equal to $-n$ times an eigenvalue of $B$ will serve the purpose. But I am not sure about general $B$. What if I take arbitrary matrices $A$ and $B$.
26315
Give an example of $2\times2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors.  I would like to know a systematic answer of how to get this. My guess was that any matrix that has 1 independent eigenvector will have the same number of independent eigenvector for the squared matrix. I have tried random selections but obviously couldn't come up with one. What's more useful will be to get the intuition behind this problem.
106119
We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the .  Theorem. Let $A$ be a positive square matrix. If any element increases in the matrix then $\lambda_{\max}$ increases.  My questions.    Is there a name for this theorem and can anybody say books or papers what refer to it? How to prove it?
90922
Let $A \in \mathbb{C}(n,n)$ and $k \geq 2$ be an integer such that $$A \sim A^k$$. Show that if $A$ is non-singular then each eigenvalue of $A$ is a root of unity.   Attempt: Since $A \sim A^k$, $$PA = A^kP$$ where $P$ is an invertible matrix. Since $A$ is invertible, $0$ cannot be an eigenvalue of $A$. Suppose $$Av = \lambda v \quad v \neq 0$$ then $$PAv = \lambda Pv$$ $$\therefore A^k(Pv) = \lambda (Pv) $$ but this implies that $Pv$ is an eigenvector of $A$. But the eigenvalues of $A^k$ are $\lambda^k$ $$\therefore \lambda^k=\lambda$$ which gives the conclusion required.   My questions is: Is the logic correct? If so, Am I missing any details? If not, then how could I approach this?   Thanks!
36772
Let $A$ be a $227\times227$ matrix with entries in $\mathbb{Z}_{227}$, such that all the eigenvalues are distinct. What is the trace of $A$?
110854
If $N$ is a $k \times k$ elementary nilpotent matrix, i.e. $N^k = 0$ but $N^{k-1} \ne 0$, then show that $N^\top$ is similar to $N$.  Now use the Jordan form to prove that every complex $n \times n$ matrix is similar to its transpose.  I have figured out the second part, and am struggling with connecting nilpotency to being similar to the transpose.
89869
Given a symmetric matrix $B \in \mathbb{C}^{n\times n}$.  How many coefficients of  $A \in \mathbb{C}^{n\times n}$ can you obtain from the following equation? $$A^\top A=B$$  I think this problem is under determined? Isn't it? Sure $B$ must be symmetric. Thus  only $\frac{n(n+1)}{2}$ complex equations remain for $n\times n$ coefficients.
61681
Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \  \lambda \in \mathbb R\}$$  ($I_n$ is the $n \times n$ identity matrix)  I don't know how to use $E_{ij}$ (matrix with $1$ in $(i,j)$ and $0$ elsewhere) and the elementary matrix $P_{ij}$ to prove this question. Can anyone explain it please?
45709
We were asked to prove the following:  Let $ A $ be an $n \times n$ matrix with $n$ distinct real eigenvalues. If $AB=BA$, show that $B$ is diagonalizable.   It was suggested I show that an eigenvector of $A$ is also an eigenvector of $B$. I am both having trouble doing this and failing to see how I would complete the proof after. Any help would be appreciated.  Thanks
25928
Suppose I have an eigenvalue $\lambda$, now I want to determine the number of Jordan blocks corresponding to that eigenvalue, as well as size of each block. I know that:  number of blocks is equal to the number of eigenvectors there are $\dim \ker (A - \lambda I)^k - \dim \ker (A - \lambda I)^{k-1}$ blocks of size at least $k$ Why exactly are those statements true. Could someone provide me with a proof or at least some reasoning behind this?
42739
Let, $A$ &amp; $B$ are $n\times n$ positive definite matrices &amp; $I$ be the $n\times n$ identity matrix. Then which of the followings are positive definite?  (a) $A+B$  (b) $ABA$  (c) $A^{2}+I$  (d) $AB$  I know that, $A^{2}+I$ is positive definite, as if $\lambda$ is an eigen value of $A$ then $(1+\lambda^2)$ is an eigen value of $A^2+I$.  I think (d) is true.Suppose, $\lambda_{1}$ &amp; $\lambda_{2}$ be two eigen values of $A_{2\times 2}$ matrix &amp; $\beta_{1}$, $\beta_{2}$ be two eigen values of $B_{2\times 2}$ matrix.  Now, $det(AB)=det(A)det(B)=\lambda_{1}.\lambda_{2}.\beta_{1}.\beta_{2}.$  As, $\lambda_{1},\lambda_{2},\beta_{1},\beta_{2}$ are all positive so eigen values of $AB$ are all positive, so $AB$ is positive definite.  Similarly, $ABA$ is positive definite.But I am not sure &amp; I have no idea about $A+B$.
108649
Let $A$ and $B$ be $m \times n$ and $n \times m$ real matrices.  I proved that if $\lambda$ is a nonzero eigenvalue of the $m \times m$ matrix $AB$ then it is also an eigenvalue of the $n \times n$ matrix $BA$. But I am having trouble finding an example showing that this need not be true if $\lambda=0$  Thank you in advance