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GeezerVison is showing mathworld.wolfram.com — Matrix Norm -- from Wolfram MathWorld

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Matrix Norm

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A matrix norm on square matrices over R or C, in the submultiplicative convention used here, assigns a nonnegative number ||A|| to each matrix A and has the properties

1. ||A||>0 when A!=0 and ||A||=0 iff A=0,

2. ||kA||=|k|||A|| for any scalar k,

3. ||A+B||<=||A||+||B||,

4. ||AB||<=||A||||B||.

Some authors require only the first three properties and call the fourth property submultiplicativity (Higham 2021). Norms satisfying the fourth property obey the following eigenvalue bound. If A is a nonsingular matrix and lambda is any of its eigenvalues, then

1/(||A^(-1)||)<=|lambda|<=||A||.
(1)

The induced matrix p-norm is also defined for an m×n matrix A, with m,n>=1, and 1<=p<=infty, by

||A||_p=max_(x s.t. |x|_p=1)|Ax|_p,
(2)

where |x|_p is a vector norm. The cases p=1, 2, and infty have the explicit formulas below. For other p, an explicit formula is not generally available (Higham 1992, pp. 539-540).

Matrix norms are implemented as Norm[m, p], where p may be 1, 2, Infinity, or "Frobenius".

The maximum absolute column sum norm ||A||_1 is defined as

||A||_1=max_(j)sum_(i=1)^m|a_(ij)|.
(3)

The spectral norm ||A||_2, which is the square root of the maximum eigenvalue of A^HA (where A^H is the conjugate transpose),

||A||_2=(maximum eigenvalue of A^HA)^(1/2)
(4)

is often referred to as "the" matrix norm.

The maximum absolute row sum norm is defined by

||A||_infty=max_(i)sum_(j=1)^n|a_(ij)|.
(5)

||A||_1, ||A||_2, and ||A||_infty satisfy the inequality

||A||_2^2<=||A||_1||A||_infty.
(6)

See also

Compatible, Frobenius Norm, Hilbert-Schmidt Norm, Maximum Absolute Column Sum Norm, Maximum Absolute Row Sum Norm, Natural Norm, Norm, Polynomial Norm, Schatten Norm, Spectral Norm, Spectral Radius, Vector Norm

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More things to try:

  • norm {{a, b}, {c, d}}
  • norm {{2,4,3},{0,1,1},{3,5,5}}.{{2,4,3},{0,1,1},{3,5,5}}
  • norm {{a,b,c},{d,e,f},{g,h,i}}

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. J. "Estimating the Matrix p-Norm." Numer. Math. 62, 539-555, 1992. https://doi.org/10.1007/BF01396242.Higham, N. J. "Matrix Norms." §6.2 in Accuracy and Stability of Numerical Algorithms. Philadelphia, PA: Soc. Industrial and Appl. Math., 1996.Higham, N. J. "What Is a Matrix Norm?" Oct. 12, 2021. https://nhigham.com/2021/10/12/what-is-a-matrix-norm/.Horn, R. A. and Johnson, C. R. "Norms for Vectors and Matrices." Ch. 5 in Matrix Analysis. Cambridge, England: Cambridge University Press, 1990.

Referenced on Wolfram|Alpha

Matrix Norm

Cite this as:

Weisstein, Eric W. "Matrix Norm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MatrixNorm.html

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