Matrix Norm
Given a square complex or real matrix 
, a matrix norm 
is a nonnegative number
associated with 
having the properties
1. 
when 
and 
iff 
,
2. 
for any scalar 
,
3. 
,
4. 
.
Let 
, ..., 
be the eigenvalues of
, then
 |
(1) |
The matrix 
-norm
is defined for a real number 
and a matrix 
by
 |
(2) |
where 
is a vector norm. The task of computing a matrix 
-norm is difficult for 
since it is a nonlinear optimization problem with constraints.
Matrix norms are implemented as Norm[m, p], where 
may be 1, 2, Infinity, or "Frobenius".
The maximum absolute column sum norm 
is defined as
 |
(3) |
The spectral norm 
, which is the square root
of the maximum eigenvalue of 
(where 
is the conjugate transpose),
 |
(4) |
is often referred to as "the" matrix norm.
The maximum absolute row sum norm is defined by
 |
(5) |
, 
, and 
satisfy the inequality
 |
(6) |
See also
Compatible, Frobenius Norm, Hilbert-Schmidt Norm, Maximum Absolute Column Sum Norm, Maximum Absolute Row Sum Norm, Natural Norm, Norm, Polynomial Norm, Spectral Norm, Spectral Radius, Vector Norm
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References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
pp. 1114-1125, 2000.Higham, N. "Estimating the Matrix 
-Norm." Numer. Math. 62,
539-555, 1992.Higham, N. J. "Matrix Norms." §6.2
in Accuracy
and Stability of Numerical Algorithms. Philadelphia: Soc. Industrial and
Appl. Math., 1996.Horn, R. A. and Johnson, C. R. "Norms
for Vectors and Matrices." Ch. 5 in Matrix
Analysis. Cambridge, England: Cambridge University Press, 1990.
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Cite this as:
Weisstein, Eric W. "Matrix Norm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MatrixNorm.html