Eigenvalues of $A+B$

Question

$A,B$ are symmetric matrices, $A$ has eigenvalues in $[a,b]$ and $B$ has eigenvalues in $[c,d]$ then we need to show that eigenvalues of $A+B$ lie in $[a+c,b+d]$, I am really not getting where to start. What I know $A,B$ have real eigenvalues, they are diagonalizable also.

Answer 1

We can use Rayleigh quotients: for a symmetric matrix $M$ and $x\neq 0$, it is defined as $R_M(x):=\frac{\langle Mx,x\rangle}{\lVert x\rVert^2}$. If $\lambda_1\leq\ldots\leq \lambda_n$ are the eigenvalues of $M$, then $$\lambda_1=\min_{x\neq 0}R_M(x)\quad\mbox{and}\quad\lambda_n=\max_{x\neq 0}R_M(x).$$ To see this, use the fact that $M$ is diagonalizable in an orthonormal basis of eigenvectors (in order to just deal with the case $M$ diagonal.

Once you have this result, the mininmal eigenvalue of $A+B$ is $\geq\min_{x\neq 0}\frac{\langle Ax,x\rangle}{\lVert x\rVert^2}+\frac{\langle Bx,x\rangle}{\lVert x\rVert^2}\geq a+c$. Use a similar argument for the $\max$.