# Symmetric matrix with $a_{ij} = 0$ for all $|i - j| > 1$ has all eigenvalues of multiplicity $1$

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.

• for part(2), let X = diag(1,2,0,3) and Y = diag(4,0,5,6). Oct 17, 2012 at 20:29
• @Inquest What if x=y=1? Oct 17, 2012 at 20:33
• In part 1. you have the Jacobi matrices. This property is well-known.
Oct 17, 2012 at 20:41
• @Timothy. Argh. You are right. Ignore my comment. Oct 17, 2012 at 20:45

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 $G(Z))$.

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!