Let a>1 and A∈Mn(R) symmetric matrix such that A−λI is Positive-definite matrix (All eigenvalues >0) for every $\lambda
First, I'm not sure what does it mean that A−λI is positive definite for every $\lambda 0andalleigenvaluesarebiggerthan0$ or it's not.
Then, If it's symmetric I can diagonalize it, I'm not sure what to do...
Thanks!
Answer
A−λI is positive definite for every $\lambda0,A-(a-\epsilon) Iispositivedefinite.Thatmeans,eacheigenvalueofAislargerthana-\epsilon,thustheirproduct\det A\ge \prod (a-\epsilon)...let\epsilon$ goes to zero, you get what you want.
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