Smallest Eigen value of Matrix + other matrix less than greatest value eigen value of Matrix

Question

I've been working on this problem for a little bit and I'm not sure if it can be proven with the given information. Any help would be greatly appreciated to either confirm or deny my suspicion.

Assume that Q is a symmetric n × n matrix, c ∈ $R^n$ is a nonzero (column) vector, and μ is a positive number. Consider the symmetric matrix R = Q + μcc$^T$ .

Using λ1 = smallest eigenvalue of R = min (${v\neq0}$) {$$\frac{v^TRv}{v^tv}$$}

Show that for n>1 $$λ1(R) ≤ λn(Q).$$ Where λn(Q) is the largest eigenvalue of Q

Thanks for any and all help!

Answer 1

Take a vector $v \ne 0$ with $c^T v = 0$. Then $R v = Q v$. Thus

$$\lambda_1(R) \le \dfrac{v^T R v}{v^T v} = \dfrac{v^T Q v}{v^T v} \le \lambda_n(Q) $$