Proof of subordinate matrix norm equality

Proof of subordinate matrix norm equality

Let cdot_v be a norm in mathbbKn and Q in M_nmathbbK a regular matrix. We define cdot_v: mathbbKn rightarrow mathbbR such that X_vQX_v. Prove that cdot_v is also a norm. Let cdot_M and cdot_M their respective subordinate matrix norms in M_nmathbbK. Prove that forall A in M_nmathbbKn it is true that A_MQAQ-1_M. My definition of subordinate matrix norm is the following: If cdot_v is a norm then its subordinate matrix norm is A_M max_Xneq0 fracAX_vX_vmax_X1 AX_vmax_Xleq1 AX_v. The first question is pretty straight forward but Im stuck in the second one. First of all, we notice that: A_M max_Xneq0 fracAX_vX_v max_Xneq0 fracQAX_vQX_v. QAQ-1_M max_Xneq0 fracQAQ-1X_vX_v. I ve tried to prove that A_M leq QAQ-1_M and A_M geq QAQ-1_M playing with this inequality that has been useful other times when proving norm inequalities: AX_v leq A_M X_v forall A in M_nmathbbK, forall X in mathbbKn. But I have failed to prove any of both inequalities.