Is the minimum of ${\left\lVert{x^TA}\right\lVert}$, where $x \in \mathbb{R}^n$, and $A^{n \times n}$ is a symmetric real matrix, related to the smallest eigenvalue of $A$? I read about Rayleigh Quotient, but this is different, and I was wondering if there are related problems, theorems, etc.
This obviously has a trivial solution; but if we have a norm constraint on $x$, would it be similar to some well-known problem?
The minimum of $||x^TA||$ is $0$ where $x=0$, but we can modify it as $${||x^TA||\over ||x||}$$to make it a very interesting problem.
Note that you can write $$A=QDQ^T$$where $D$ is diagonal and $Q$ is unitary. Therefore $$||x^TA||{=x^TAA^Tx\\=x^TQDQ^TQD^TQ^Tx\\=x^TQDD^TQ^Tx}$$by defining $y\triangleq Q^Tx$ we have$$||x^TA||=||y^TD||\\||y||=||x||$$therefore $${||x^TA||\over ||x||}={||y^TD||\over ||y||}$$and $$\min_x{||x^TA||\over ||x||}{=\min_{y}{||y^TD||\over ||y||}\\=\min_{||y||=1}{||y^TD||}\\=\min_{i} \sigma_i}$$which is the smallest singular value of $A$. Similarly$$\max_x{||x^TA||\over ||x||}=\max_{i} \sigma_i$$