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Matrix Norm


Given a square complex or real matrix A, a matrix norm ||A|| is a nonnegative number associated with A having 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||.

Let lambda_1, ..., lambda_n be the eigenvalues of A, then

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

The matrix p-norm is defined for a real number 1<=p<=infty and a matrix A by

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

where |x|_p is a vector norm. The task of computing a matrix p-norm is difficult for p>1 since it is a nonlinear optimization problem with constraints.

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)^n|a_(ij)|.
(3)

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

 ||A||_2=(maximum eigenvalue of A^(H)A)^(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, 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 p-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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Matrix Norm

Cite this as:

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

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