# Problems on Symmetric Matrices

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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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
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