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The question is: Prove/disprove $||A^2|| \leq ||A||^2$ where A is some nxn matrix.

I've played around with a while few matrices and I'm pretty sure that this is correct but I can't quite figure out how to prove it. My first guess is just define some arbitrary matrix A, and show that for any vector $x$ of size 1, $||A^2 x|| \leq ||Ax||^2$ however this method seems rather involved.

Any tips would be great! :)

Edit: $||A||= \max ||Ax||$, where $|x| = 1$

Edit: Wow this was rather fast. Thanks everyone.

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  • $\begingroup$ Knowing about eigenvalues maybe provides a shortcut. Have you done eigenvalues yet? $\endgroup$ – mathreadler Jan 21 '16 at 1:25
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Recall that by the definition of $\|A\|$ we have $$\|Ay\| \leq \|A\|\|y\|.$$

For any $x \in \mathbb{R}^n$, using this inequality twice (on $y = Ax$ and on $y = x$), we have $$\|A^2x\| = \|A(Ax)\| \leq \|A\| \|Ax\| \leq \|A\| \left( \|A\| \|x\| \right) = \|A\|^2 \|x\|.$$

Therefore $$\|A^2\| = \sup_{x \neq 0}\dfrac{\|A^2x\|}{\|x\|} \leq \|A\|^2.$$

Note that what the matrix norm is does not matter in the proof, as long as the norm is an induced norm.

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It is true. In fact, for all $x \in \mathbb{R}^n$ with $\|x\| = 1$. If $Ax \neq 0$, then by definition, $$\|A^2x\| = \|A(Ax)\| \leq \|Ax\| \|A\|\frac{\|x\|}{\|Ax\|} = \|A\|\|x\| = \|A\|.$$ The above inequality also holds for $x$ such that $Ax = 0$. Thus $$\|A^2\| = \sup_{\|x\| = 1} \|A^2 x \| \leq \|A\|.$$

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