Spread of a matrix: Difference between revisions

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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.

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+{{Short description|Mathematical term}}
-{{Context|date=October 2009}}
+{{One source|date=July 2022}}
-In [[Matrix (mathematics)|matrix theory]], the '''spread of a matrix''' describes how far apart the eigenvalues are in the [[complex plane]].
+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>.  Then the '''spread''' of <math>A</math> is the [[non-negative number]]
+==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>
@@ Line 8: / Line 13: @@
 ==Examples==
-*For the [[zero matrix]] and the [[identity matrix]], the spread is zero.
+*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.
-*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>.
+*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 an [[unitary matrix]] <math>A</math> lie on the [[unit circle]]. Hence <math>s(A)\le 2</math>.
+*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, so if <math>B</math> is [[invertible matrix|invertible]], then
+*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>.
-::<math> s(A) = s(BAB^{-1}).</math>
 ==See also==
@@ Line 19: / Line 22: @@
 ==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.
+* [[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}}