Generalized induced norms

S. Hejazian; M. Mirzavaziri; Mohammad Sal Moslehian

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 1, page 127-133
  • ISSN: 0011-4642

Abstract

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Let ∥ · ∥ be a norm on the algebra ℳ n of all n × n matrices over ℂ . An interesting problem in matrix theory is that “Are there two norms ∥ · ∥ 1 and ∥ · ∥ 2 on ℂ n such that ∥ A ∥ = max { ∥ A x ∥ 2 ∥ x ∥ 1 = 1 } for all A ∈ ℳ n ?” We will investigate this problem and its various aspects and will discuss some conditions under which ∥ · ∥ 1 = ∥ · ∥ 2 .

How to cite

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Hejazian, S., Mirzavaziri, M., and Moslehian, Mohammad Sal. "Generalized induced norms." Czechoslovak Mathematical Journal 57.1 (2007): 127-133. <http://eudml.org/doc/31118>.

@article{Hejazian2007,
abstract = {Let $\Vert \{\cdot \}\Vert $ be a norm on the algebra $\{\mathcal \{M\}\}_n$ of all $n\times n$ matrices over $\{\mathbb \{C\}\}$. An interesting problem in matrix theory is that “Are there two norms $\Vert \{\cdot \}\Vert _1$ and $\Vert \{\cdot \}\Vert _2$ on $\{\mathbb \{C\}\}^n$ such that $\Vert A\Vert =\max \lbrace \Vert Ax\Vert _\{2\}\: \Vert x\Vert _\{1\}=1\rbrace $ for all $A\in \{\mathcal \{M\}\}_n$?” We will investigate this problem and its various aspects and will discuss some conditions under which $\Vert \{\cdot \}\Vert _1=\Vert \{\cdot \}\Vert _2$.},
author = {Hejazian, S., Mirzavaziri, M., Moslehian, Mohammad Sal},
journal = {Czechoslovak Mathematical Journal},
keywords = {induced norm; generalized induced norm; algebra norm; the full matrix algebra; unitarily invariant; generalized induced congruent; induced norm; algebra norm; full matrix algebra},
language = {eng},
number = {1},
pages = {127-133},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized induced norms},
url = {http://eudml.org/doc/31118},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Hejazian, S.
AU - Mirzavaziri, M.
AU - Moslehian, Mohammad Sal
TI - Generalized induced norms
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 127
EP - 133
AB - Let $\Vert {\cdot }\Vert $ be a norm on the algebra ${\mathcal {M}}_n$ of all $n\times n$ matrices over ${\mathbb {C}}$. An interesting problem in matrix theory is that “Are there two norms $\Vert {\cdot }\Vert _1$ and $\Vert {\cdot }\Vert _2$ on ${\mathbb {C}}^n$ such that $\Vert A\Vert =\max \lbrace \Vert Ax\Vert _{2}\: \Vert x\Vert _{1}=1\rbrace $ for all $A\in {\mathcal {M}}_n$?” We will investigate this problem and its various aspects and will discuss some conditions under which $\Vert {\cdot }\Vert _1=\Vert {\cdot }\Vert _2$.
LA - eng
KW - induced norm; generalized induced norm; algebra norm; the full matrix algebra; unitarily invariant; generalized induced congruent; induced norm; algebra norm; full matrix algebra
UR - http://eudml.org/doc/31118
ER -

References

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  1. Matrix Norms and Their Applications. Operator Theory: Advances and Applications,  36, Birkhäuser-Verlag, Basel, 1988. (1988) MR1015711
  2. Matrix Analysis. Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997. (1997) MR1477662
  3. Matrix Analysis, Cambridge University Press, Cambridge, 1994. (1994) MR1084815
  4. Norm hull of vectors and matrices, Linear Algebra Appl. 257 (1997), 1–27. (1997) MR1441701
  5. Real and Complex Analysis, McGraw-Hill, New York, 1987. (1987) Zbl0925.00005MR0924157

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