Spread of a matrix

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.


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