Monotonicity properties of norms. (English) Zbl 0717.15015

Let \({\mathbb{C}}^ d\) denote complex d space, i.e., the set of \(d\times 1\) complex matrices. Let n be a norm on \({\mathbb{C}}^ d\) (n: \({\mathbb{C}}^ d\to {\mathbb{R}}\) has n(x)\(\geq 0\) with equality only if \(x=0\), \(n(\alpha x)=| \alpha | n(x)\), and \(n(x+y)\leq n(x)+n(y)\) for all \(x,y\in {\mathbb{C}}^ d\), \(\alpha\in {\mathbb{C}})\). The notations \(| \cdot |\) and \(\leq\) on \({\mathbb{C}}^ d\) are taken componentwise. n is absolute if \(n(x)=n(| x|)\) for all \(x\in {\mathbb{C}}^ d\); n is monotone if n(x)\(\leq n(y)\) for all \(x,y\in {\mathbb{C}}^ d\) with \(| x| \leq | y|\). Several properties involving these concepts are considered for that of a particular norm n (which n may or may not satisfy). A complete set of implications among the properties is given.
Reviewer: R.Sinkhorn


15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
Full Text: DOI


[1] Bauer, F.L.; Stoer, J.; Witzgall, C., Absolute and monotonic norms, Numer. math., 3, 257-264, (1961) · Zbl 0111.01602
[2] Deutsch, E., Solution of advanced problem 6249, Amer. math. monthly, 87, 831, (1980)
[3] Funderlic, R.E., Some characterizations of orthant monotonic norms, Linear algebra appl., 28, 77-83, (1979) · Zbl 0419.15001
[4] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge U. P · Zbl 0576.15001
[5] R. A. Horn and R. Mathias, An analog of the Cauchy-Schwarz inequality for Hadamard products and unitarily invariant norms, SIAM J. Matrix Anal. Appl., to appear. · Zbl 0722.15019
[6] Johnson, C.R., Two submatrix properties of certain induced norms, J. res. nat. bur. standards, 79B, 97-101, (1975) · Zbl 0333.15014
[7] Maítre, J.F., Sur certaines normes et fonctionelles dans LES espaces de matrices et d’operateurs, ()
[8] Malek-Shamirzadi, M., On monotonic and orthant monotonic norms, Linear algebra appl., 56, 169-175, (1984) · Zbl 0525.15018
[9] Merikoski, J.K., On operator norms of submatrices, Linear algebra appl., 36, 173-183, (1981) · Zbl 0455.15024
[10] Merikoski, J.K., Some notes on absolute norms, () · Zbl 0467.15013
[11] Okubo, K., Hölder type norm inequalities for Schur products of matrices, Linear algebra appl., 91, 13-28, (1987) · Zbl 0633.15013
[12] Stoer, J., A characterization of Hölder norms, J. soc. indust. appl. math., 12, 634-648, (1964) · Zbl 0196.29803
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.