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2009-10-16T05:17:14Z

Replace Category:Articles lacking sources (Erik9bot) with unref tag using AWB

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{{Context|date=October 2009}}
In [[Matrix (mathematics)|matrix theory]], the '''spread of a matrix''' describes how far apart the eigenvalues are in the [[complex plane]].

Suppose <math>A</math> is a [[square matrix]] with [[eigenvalues]] <math>\lambda_1, \ldots, \lambda_n</math>. Then the '''spread''' of <math>A</math> is the [[non-negative number]]

:<math>s(A) = \max \{|\lambda_i - \lambda_j| : i,j=1,\ldots n\}.</math>

==Examples==

*For the [[zero matrix]] and the [[identity matrix]], the spread is zero.
*Only <math>0</math> and <math>1</math> can be eigenvalues for a [[projection (mathematics)|projection]]. A [[projection matrix]] therefore has spread <math>0</math> or <math>1</math>.
*All eigenvalues of an [[unitary matrix]] <math>A</math> lie on the [[unit circle]]. Hence <math>s(A)\le 2</math>.
*The spread of a matrix depends only on the [[spectral theory|spectrum]] of the matrix, so if <math>B</math> is [[invertible matrix|invertible]], then

::<math> s(A) = s(BAB^{-1}).</math>

==See also==
* [[Field of values]]

==References==
* Marvin Marcus and Henryk Minc, ''A survey of matrix theory and matrix inequalities'', [[Dover Publications]], 1992, ISBN 0-486-67102-X. Chap.III.4.

[[Category:Linear algebra]]
[[Category:Matrix theory]]

{{Linear-algebra-stub}}

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en>Rich Farmbrough: Replace :Category:Articles lacking sources (Erik9bot) with unref tag using AWB