For any real symmetric matrix X and Q with the appropriate dimension, does the following inequality relation holds?
λmin(Q) ‖X‖2 ≤ X^TQX ≤ λmax(Q) ‖X‖2
where λmin(Q) and λmax(Q) represent the minimum and maximum of matrix Q eigenvalues, respectively, and ‖X‖2 represents the 2-norm of the matrix X.
Yes, I think that if it is true, there is more than one theorem that could prove such an inequality, but to do so I would have to work on it and now I do not have all the time available, but you can write more details about this problem in a more finished document and Sure, send it to my email.
Excuse me, what sense should be given to the inequality? In the way it is written, the central term is a matrix, while the extreme terms are scalar.
I assume by writing A ≥ (scalar) you mean that the matrix A-(scalar)*(Identity) is positive semi-definite. I also assume the ||X||_2 should be squared on both sides for the inequality to make sense. The upper bound is true because it follows trivially from the operator norm inequality ||AB|| ≤ ||A|| ||B|| and we also have ||A|| ≤ ||A||_2. However the lower bound is false because one could take X = diag(0,1) and Q = diag(1,1) = Identity.
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