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Keywords:
induced norm; generalized induced norm; algebra norm; the full matrix algebra; unitarily invariant; generalized induced congruent
induced norm; generalized induced norm; algebra norm; the full matrix algebra; unitarily invariant; generalized induced congruent
Summary:
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 $\Vert {\cdot }\Vert _1=\Vert {\cdot }\Vert _2$.
References:
[1] G. R. Belitskiĭ, Yu. I. Lyubich: Matrix Norms and Their Applications. Operator Theory: Advances and Applications, 36. Birkhäuser-Verlag, Basel, 1988. MR 1015711
[2] R. Bhatia: Matrix Analysis. Graduate Texts in Mathematics, 169. Springer-Verlag, New York, 1997. MR 1477662
[3] R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 1994. MR 1084815
[4] C.-K. Li, N.-K. Tsing, and F. Zhang: Norm hull of vectors and matrices. Linear Algebra Appl. 257 (1997), 1–27. MR 1441701
[5] W. Rudin: Real and Complex Analysis. McGraw-Hill, New York, 1987. MR 0924157 | Zbl 0925.00005
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