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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.

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  • $\begingroup$ for part(2), let X = diag(1,2,0,3) and Y = diag(4,0,5,6). $\endgroup$
    – Inquest
    Oct 17, 2012 at 20:29
  • $\begingroup$ @Inquest What if x=y=1? $\endgroup$
    – Ester
    Oct 17, 2012 at 20:33
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    $\begingroup$ In part 1. you have the Jacobi matrices. This property is well-known. $\endgroup$
    – PAD
    Oct 17, 2012 at 20:41
  • $\begingroup$ @Timothy. Argh. You are right. Ignore my comment. $\endgroup$
    – Inquest
    Oct 17, 2012 at 20:45
  • $\begingroup$ Can anyone please help me with the second one ? $\endgroup$
    – Ester
    Oct 17, 2012 at 21:20

2 Answers 2

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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 $G(Z))$.

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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!
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