Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

1 . Let $A = (a_{ij})$ be a real $n \times n$ matrix such that $a_{ij} = a_{ji}$ for all $1 \leq i,j \leq n$ and $a_{ij} = 0$ for $|i-j|>1$. Moreover $a_{ij}$ is non-zero for all $i$,$j$ satisfying $|i-j| = 1$. Show that all the eigenvalues of $A$ are of multiplicity $1$.

2 . Give examples of 2 real $n \times n$ matrices $X = (x_{ij})$, $Y = (y_{ij})$ where $x_{ij} = x_{ji}$ and $y_{ij} = y_{ji}$ for all $1 \leq i,j \leq n$ so that $xX +yY$ has $n$ non-repeated eigenvalues for all real numbers $x$, $y$ where $x$, $y$ are not zero simultaneously.

Thank you for any help.

share|improve this question
    
for part(2), let X = diag(1,2,0,3) and Y = diag(4,0,5,6). –  Inquest Oct 17 '12 at 20:29
    
@Inquest What if x=y=1? –  Ester Oct 17 '12 at 20:33
1  
In part 1. you have the Jacobi matrices. This property is well-known. –  PAD Oct 17 '12 at 20:41
    
@Timothy. Argh. You are right. Ignore my comment. –  Inquest Oct 17 '12 at 20:45
    
Can anyone please help me with the second one ? –  Ester Oct 17 '12 at 21:20

2 Answers 2

2) Let $X,Y$ be linear independent real symmetric matrices of order 2 and trace zero.

Let $Z$ be any linear combination of X and Y. Notice that $Z$ has the same properties. Therefore its eigenvalues $(Z$ is diagonalizable since it is real symmetric$)$ have oppositive signs $($their sum must be zero$)$, unless $Z$ is the zero matrix. But it occurs only if $Z$ is the trivial combination of $X$ and $Y$.

Now for matrices of order $2k$ $(2k+1)$, instead of $X$ and $Y$, use $F(X)$ and $F(Y)$ $(G(X)$ and $G(Y))$.

$F(X)=\left(\begin{array}{cccc} X & 0 & \dots & 0 \\ 0 & 2X & \dots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & 0 & kX \end{array} \right)_{2k\times2k}$ $G(X)=\left(\begin{array}{cc} F(X) & 0_{2k\times 1} \\ 0_{1\times 2k} & 0_{1\times 1} \end{array} \right)_{2k+1\times 2k+1}$

If $Z$ is any linear combination of $X$ and $Y$ then $F(Z)$ $(G(Z))$ is be the respective linear combination of $F(X)$ and $F(Y)$ $(G(X)$ and $G(Y))$.

If $a,-a$ are the eigenvalues of $Z$ then $a,-a,2a,-2a,\dots,ka,-ka$ are the eigenvalues of $F(Z)$ $(a,-a,2a,-2a,\dots,ka,-ka,0$ are the eingevalues of $G(Z))$.

share|improve this answer

Hints.

  1. Let $\lambda$ be an eigenvalue of $A$. As $A$ is diagonalisable, can you relate the geometric multiplicity of $\lambda$ to the rank of $\lambda I-A$? Now, let $B$ be the submatrix obtained by deleting the first row and last column of $\lambda I-A$. What is the rank of $B$? Then, what is the rank of $\lambda I-A$?
  2. Split a matrix in the form of $A$ in part 1 into two appropriate symmetric matrices!
share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.