Spread of a matrix: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 23:37, 11 March 2024

In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix.

Definition

Let $A$ be a square matrix with eigenvalues $\lambda _{1},\ldots ,\lambda _{n}$ . That is, these values $\lambda _{i}$ are the complex numbers such that there exists a vector $v_{i}$ on which $A$ acts by scalar multiplication:

Av_{i}=\lambda _{i}v_{i}.

Then the spread of $A$ is the non-negative number

s(A)=\max\{|\lambda _{i}-\lambda _{j}|:i,j=1,\ldots n\}.

Examples

For the zero matrix and the identity matrix, the spread is zero. The zero matrix has only zero as its eigenvalues, and the identity matrix has only one as its eigenvalues. In both cases, all eigenvalues are equal, so no two eigenvalues can be at nonzero distance from each other.
For a projection, the only eigenvalues are zero and one. A projection matrix therefore has a spread that is either $0$ (if all eigenvalues are equal) or $1$ (if there are two different eigenvalues).
All eigenvalues of a unitary matrix $A$ lie on the unit circle. Therefore, in this case, the spread is at most equal to the diameter of the circle, the number 2.
The spread of a matrix depends only on the spectrum of the matrix (its multiset of eigenvalues). If a second matrix $B$ of the same size is invertible, then $BAB^{-1}$ has the same spectrum as $A$ . Therefore, it also has the same spread as $A$ .

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.

@@ Line 1: / Line 1: @@
+{{Short description|Mathematical term}}
-In [[matrix theory]], the spread of a matrix describes how far apart the eigenvalues are in the [[complex plane]].
+{{One source|date=July 2022}}
+In [[mathematics]], and more specifically [[Matrix (mathematics)|matrix theory]], the '''spread of a matrix''' is the largest distance in the [[complex plane]] between any two [[eigenvalue]]s of the matrix.
-Suppose <math>A</math> is a [[square matrix]] with [[eigenvalues]]
-<math>\lambda_1, \ldots, \lambda_n</math>.
+==Definition==
+Let <math>A</math> be a [[square matrix]] with eigenvalues <math>\lambda_1, \ldots, \lambda_n</math>. That is, these values <math>\lambda_i</math> are the [[complex number]]s such that there exists a vector <math>v_i</math> on which <math>A</math> acts by [[scalar multiplication]]:
+:<math>Av_i=\lambda_i v_i.</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 sphere. 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>
+*For the [[zero matrix]] and the [[identity matrix]], the spread is zero. The zero matrix has only zero as its eigenvalues, and the identity matrix has only one as its eigenvalues. In both cases, all eigenvalues are equal, so no two eigenvalues can be at nonzero distance from each other.
+*For a [[projection (mathematics)|projection]], the only eigenvalues are zero and one. A [[projection matrix]] therefore has a spread that is either <math>0</math> (if all eigenvalues are equal) or <math>1</math> (if there are two different eigenvalues).
+*All eigenvalues of a [[unitary matrix]] <math>A</math> lie on the [[unit circle]]. Therefore, in this case, the spread is at most equal to the [[diameter]] of the circle, the number&nbsp;2.
+*The spread of a matrix depends only on the [[spectral theory|spectrum]] of the matrix (its multiset of eigenvalues). If a second matrix <math>B</math> of the same size is [[invertible matrix|invertible]], then <math>BAB^{-1}</math> has the same spectrum as <math>A</math>. Therefore, it also has the same spread as <math>A</math>.
+==See also==
-{{maths-stub}}
+* [[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|*]]
+[[Category:Matrix theory]]