Given is symmetric block matrices $M$ of the form
$$
M=\left(\begin{array}{cc}
A & B \\
B^{\top} & C
\end{array}\right)
$$
where $A$ is a constant, $B$ is $1 \times n$ vector and $C$ is a $n \times n$ symmetric matrix. Under which conditions on eigenvalues of $C$ is this matrix negative definite? Can I say if matrix C is negative definite, M will be negative definite too since A is a constant?