# On matrix norms

It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:

$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.

Suppose we define a different function of matrices this way:

$f(A)=\inf_{x \neq 0}{\frac{||Ax||}{||x||}}$.

Has $f(\cdot)$ been studied before? Does it have a standard name?

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$|||A|||$ is the largest s-number (modulus of gen. eigenvalue). $f(A)$ is the smallest s-number. It is 0 if $A$ is not injective.

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S-numbers are also known as singular values. – Federico Poloni Feb 9 '13 at 16:07
This holds if the underlying norm is the Euclidean norm. With other norms, I have no idea if this has ever been studied. – Federico Poloni Feb 9 '13 at 16:07
@ Federico: They have been treated for operators on Banach spaces. See papers and books by Albrecht Pietsch, e.g. – Peter Michor Feb 9 '13 at 16:16
@PeterMichor: Can you give a specific reference, please? Thanks! – Felix Goldberg Feb 9 '13 at 18:14
@ Felix Goldberg: MR0519680 (81a:47002) Reviewed Pietsch, Albrecht Operator ideals. Mathematische Monographien [Mathematical Monographs], 16. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. 451 pp. Newer: MR1863699 (2003h:47137) – Peter Michor Feb 9 '13 at 22:31

By the Courant-Fischer min-max theorem, if $A$ is Hermitian, then $f(A) = \lambda_n(A)$, the smallest eigenvalue of $A$.

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Right! Of course, I managed to utterly confuse myself here. – Felix Goldberg Feb 9 '13 at 15:59