Let $a>1$ and $A\in M_n(\mathbb{R})$ symmetric matrix such that $A-\lambda I$ is Positive-definite matrix (All eigenvalues $> 0$) for every $\lambda
First, I'm not sure what does it mean that $A-\lambda I$ is positive definite for every $\lambda 0$ and all eigenvalues are bigger than $0$ or it's not. Then, If it's symmetric I can diagonalize it, I'm not sure what to do... Thanks!
$A\in M_n(\mathbb{R})$ symmetric matrix , $A-\lambda I$ is Positive-definite matrix- prove: $\det A\geq a^n $
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2Hint: How do the eigenvalues of $A-\lambda I$ relate to those of $A$? So, what is the relationship between $a$ and the eigenvalues of $A$? – 2011-08-07
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2You seem to be confused in the second paragraph. You are given that $A-\lambda I$ (and **not** the matrix $A$) is positive definite for all $\lambda < a$, which makes perfect sense. – 2011-08-07
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0Thank you for the comment. I assume that if $\alpha$ is an eigenvalue of A so $\alpha - \lambda$ is an eigenvalue of $A-\lambda I$. You right regarding the confusion, I'll fix it. But, what does it mean to be positive define for onlt $\lambda < a$? – 2011-08-07
3 Answers
$A-\lambda I$ is positive definite for every $\lambda0$, $A-(a-\epsilon) I$ is positive definite. That means, each eigenvalue of $A$ is larger than $a-\epsilon$, thus their product $\det A\ge \prod (a-\epsilon)$ ... let $\epsilon$ goes to zero, you get what you want.
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0Since eigenvalues are fixed, you could also simply say they are all${}\geq a$ and then multiply them. – 2013-05-26
Since $A$ is symmetric it is orthogonally diagonalizable and since orthogonal similarity preserves positive definitness, we may assume for all intents and purposes that $A$ is diagonal to begin with. We then know that $a_{ii} - \lambda > 0$ for all $\lambda < a$ and $1 \le i \le n$, that is $a_{ii} > \lambda$ for all $\lambda < a$ and for all $i$. It follows that $a_{ii} \ge a$ for all $i$, hence $\det A = \prod_{i=1}^{n}a_{ii} \ge a^n$.
I think you should not use $\lambda$ in $A-\lambda I$ because $\lambda$ usually denotes eigenvalues. So using $A-kI$ for example may be less confusing.
Let $\mu_i(A)$ be the eigenvalues of $A$. Then the eigenvalues of $A-kI$ simply are $\mu_i(A)-k$. Since $A-kI$ is p.d., we have $\mu_i(A)>k, \forall k
EDIT: I'm not sure whether we need use limitation to solve the problem as Sunni does.