Let $A \in M\_n(\mathbb{R})$ be a **symmetric** matrix and assume we perform on $A$ an elementary row operation $A \xrightarrow{R\_i = R\_i + cR\_j} B$ where $i \neq j$ and $c \in \mathbb{R}$ to get $B$. Are all the eigenvalues of $B$ necessarily real?




---


This might look like a strange question to ask so here is some motivation. Let's say the signature of an arbitrary matrix $A \in M\_n(\mathbb{R})$ is $(p,q,r)$ if $p$ is the number of eigenvalues of $A$ (counted with algebraic multiplicity) lying in the positive open half-plane, $q$ is the number of eigenvalues lying in the negative open half-plane and $r$ the number of eigenvalues of $A$ lying on the imaginary axis. For symmetric matrices, this definition coincides with the standard definition. Thinking about [this question](https://math.stackexchange.com/questions/4106402/did-i-just-discover-a-new-way-to-calculate-the-signature-of-a-matrix), I've came to suspect that if we start with a **symmetric** matrix $A$ and perform an elementary operation of the form $R\_i = R\_i + cR\_j$ to get $B$, then $B$ will have the same signature as $A$.


Thinking about this geometrically, we have a one-parameter family of matrices $A(c)$ which are obtained from $A$ by performing the operation $R\_i = R\_i + cR\_j$. In order for the signature of $A(c)$ to change as we vary $c$, we could have two positive eigenvalues of $A$ which collide, split off into a pair of conjugate eigenvalues which then "travel" across the imaginary axis.
When $A$ is not symmetric but still has real eigenvalues, this behaviour is indeed possible as the following example shows:


$$ A = \begin{bmatrix} -1 & 1 \\ 0 & -1 \end{bmatrix}, \,\,\, A(c) = \begin{bmatrix} -1 & 1 \\ -c & c - 1 \end{bmatrix}. $$


However, starting with a symmetric $A$ I couldn't find a single example where the eigenvalues of $A(c)$ are complex and not real.



Are the eigenvalues of a symmetric matrix stay real after performing a row-addition operation?
No. Example
$$A=\left(\begin{array}{cc}-2&1\\ 1&-2\end{array}\right),\quad c=1.$$


