Proving that a tridiagonal is similar to a symmetric matrix via a real diagonal matrix Let $A \in M_n(\mathbb{R})$ be a tridiagonal matrix ($a_{ij} = 0$ if $|i-j|>1$). Let $a_{i,i+1}a_{i+1,i} > 0, i=1,...,n-1$. Prove that there's a non-singular real diagonal matrix $D$ such that $D^{-1}AD$ is symmetric.
My first thought was to just pick an arbitrary diagonal matrix $D$, then try and set $D^{-1}AD$ equal to some symmetric matrix. But the calculations got a little out of hand, so I'm wondering if there's something simpler
Beyond what I attempted, I'm honestly not even sure how to get this off the ground, so if anyone can give me a nudge (or possibly a hefty shove) in the right direction, I'd really appreciate it.
Thanks in advance.
 A: A hefty shove: your approach works. If the diagonal matrix is $D:=\text{diag}(\lambda_1,\ldots,\lambda_n)$, then calculate the $i,j$ element of the product $C:=D^{-1}AD$:
$$
c_{i,j}=(D^{-1}AD)_{i,j}=\frac1{\lambda_i}a_{i,j}\lambda_j$$
The requirement that $C$ be symmetric asserts that
$c_{i,j}=c_{j,i}$ for all $i,j$. But this requirement isn't as onerous as it looks. Convince yourself that there are really only $n-1$ constraints here, since $c_{i,j}$ is mostly zero, and the case $i=j$ is automatically true. So set $\lambda_1:=1$ and use your constraints to solve for the remaining $\lambda$'s.
A: Note that any diagonal similarity can be performed by successively multiplying the $i$th column and dividing the $i$th row by $d_i$ for some $d_i$.
With that in mind, let's try a small example.
$$
A = \pmatrix{
a_{11} & a_{12} \\
a_{21} & a_{22} & a_{23}\\
&a_{32} & a_{33}}
$$
Suppose for now that all entries are positive. Taking $d_1 = \sqrt{a_{12}/a_{21}}$ gets us
$$
\pmatrix{
a_{11} & \sqrt{a_{12}a_{21}} \\
\sqrt{a_{12}a_{21}} & a_{22} & a_{23}\\
&a_{32} & a_{33}}
$$
Taking $d_3 = \sqrt{a_{32}/a_{23}}$ gives us
$$
\pmatrix{
a_{11} & \sqrt{a_{12}a_{21}} \\
\sqrt{a_{12}a_{21}} & a_{22} & \sqrt{a_{23}a_{32}}\\
&\sqrt{a_{23}a_{32}} & a_{33}}.
$$
Now, try to extend this idea to the general case.
