Spread of a matrix

In matrix theory, the spread of a matrix describes how far apart the eigenvalues are in the complex plane.

Suppose $A$ is a square matrix with eigenvalues $lambda\_1,\; ldots,\; lambda\_n$. 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.
*Only $0$ and $1$ can be eigenvalues for a projection. A projection matrix therefore has spread $0$ or $1$.
*All eigenvalues of an unitary matrix $A$ lie on the unit circle. Hence $s(A)le\; 2$.
*The spread of a matrix depends only on the spectrum of the matrix, so if $B$ is invertible, then

::$s(A)\; =\; s(BAB^\{-1\}).$

ee 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.