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If I have a matrix $A$ and vector $x$ is there such a relationship or something similar involving determinants? $$\|Ax\| \leq |\det A|\|x\|$$ where the absolute values indicate the usual Euclidean norm?

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    A bit confusing. Use $\|x\|$ (`\|x\|`) to denote a norm, and $|x|$ (`|x|`) to denote the absolute value.2012-04-20
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    Look up [operator norm](http://en.wikipedia.org/wiki/Operator_norm) for the thing you need to replace $\det$ with to make this true.2012-04-20
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    The determinant should be replaced by something like the [spectral radius](http://en.wikipedia.org/wiki/Spectral_radius).2012-04-20
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    Thanks. So can I write: $\|Ax\|_{R^n} \leq \| A \|_{R} \|x\| \leq C\|A\|_{\infty}\|x\|$ where $\|\cdot \|_{R}$ is some norm on $\mathbb{R}^n$ and $\|A\|_{\infty}$ is the biggest absolute-valued element in the matrix? I got this by equivalence of norms and so I need a constant $C$.2012-04-20

3 Answers 3

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No. Because if $\det A=0$ then $|\det A|\|x\|=0,$ for any $x$, while $\|Ax\|$ can take any value varying $x$.
Consider for example $$A=\left(\begin{array}{cc} 1 & 0\\ 0 & 0 \end{array} \right).$$

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    What if $detA\not=0$?2017-05-27
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AGTortorella showed that it cannot be true when $det(A) = 0$, but for $det(A) \ne 0$ this inequality also doesn't hold. Consider:

$A=\left(\begin{array}{cc} 1 & a\\ 0 & 1 \end{array} \right)$

$det(A) = 1$ for every $a$

Let $x = \left(\begin{array}{cc} 1 \\ 1 \end{array} \right)$

Then: $Ax = \left(\begin{array}{cc} 1 \cdot 1 + a \cdot 1 \\ 0 \cdot 1 + 1 \cdot 1 \end{array} \right) = \left(\begin{array}{cc} a + 1 \\ 1 \end{array} \right)$

So clearly:

$\|Ax\| \to \infty$ when $a \to \infty$

but

$|det(A)|\|x\|=1 \cdot \sqrt 2$ for every $a$.

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If $A$ is an invertible operator of an n dimensional vector space to itself then

$$\frac{1}{det(A^{-1})^{1/n}} \leq |A|\leq \frac{|A^{-1}|^{n-1}}{det(A^{-1})}$$

This you can get by the relation

$$s_1...s_n = det(A^{-1})$$

where $s_i$ are singular values of $A^{-1}$ which satisfy

$$\frac{1}{|A|}=s_1 \leq ... \leq s_n = |A^{-1}|$$

And you get the relation immediately.