Eigenvalue Sensitivity

For an [Graphics:../Images/tricks_gr_252.gif] matrix [Graphics:../Images/tricks_gr_253.gif] with elements [Graphics:../Images/tricks_gr_254.gif], eigenvalues [Graphics:../Images/tricks_gr_255.gif], (right) eigenvectors [Graphics:../Images/tricks_gr_256.gif] satisfying [Graphics:../Images/tricks_gr_257.gif], and left eigenvectors [Graphics:../Images/tricks_gr_258.gif] satisfying [Graphics:../Images/tricks_gr_259.gif], it is easy to obtain an expression for the sensitivity of [Graphics:../Images/tricks_gr_260.gif] to small changes in the element [Graphics:../Images/tricks_gr_261.gif].  (See [Horn and Johnson, 1990] and http://www.astro.virginia.edu/~eww6n/math/):

[Graphics:../Images/tricks_gr_262.gif]

so

[Graphics:../Images/tricks_gr_263.gif]

where [Graphics:../Images/tricks_gr_264.gif] and [Graphics:../Images/tricks_gr_265.gif] are the [Graphics:../Images/tricks_gr_266.gif]-th and [Graphics:../Images/tricks_gr_267.gif]-th components of [Graphics:../Images/tricks_gr_268.gif] and [Graphics:../Images/tricks_gr_269.gif].

The random complex matrix

[Graphics:../Images/tricks_gr_270.gif]
[Graphics:../Images/tricks_gr_271.gif]
[Graphics:../Images/tricks_gr_272.gif]

has eigenvalues

[Graphics:../Images/tricks_gr_273.gif]
[Graphics:../Images/tricks_gr_274.gif]

and (right) eigenvectors

[Graphics:../Images/tricks_gr_275.gif]
[Graphics:../Images/tricks_gr_276.gif]

The left eigenvectors are computed using [Graphics:../Images/tricks_gr_277.gif] since [Graphics:../Images/tricks_gr_278.gif] if  [Graphics:../Images/tricks_gr_279.gif].

[Graphics:../Images/tricks_gr_280.gif]
[Graphics:../Images/tricks_gr_281.gif]

Introducing the diagonal matrix  of eigenvalues,

[Graphics:../Images/tricks_gr_282.gif]
[Graphics:../Images/tricks_gr_283.gif]

we see that [Graphics:../Images/tricks_gr_284.gif],

[Graphics:../Images/tricks_gr_285.gif]
[Graphics:../Images/tricks_gr_286.gif]

and [Graphics:../Images/tricks_gr_287.gif]:

[Graphics:../Images/tricks_gr_288.gif]
[Graphics:../Images/tricks_gr_289.gif]

Moreover, the dot products of all possible pairs of left and right eigenvectors forms a matrix [Graphics:../Images/tricks_gr_290.gif], which is diagonal (since [Graphics:../Images/tricks_gr_291.gif]).

[Graphics:../Images/tricks_gr_292.gif]
[Graphics:../Images/tricks_gr_293.gif]

The diagonal entries are the products [Graphics:../Images/tricks_gr_294.gif].

[Graphics:../Images/tricks_gr_295.gif]
[Graphics:../Images/tricks_gr_296.gif]

If we perturb [Graphics:../Images/tricks_gr_297.gif]by 0.01+0.02i,

[Graphics:../Images/tricks_gr_298.gif]
[Graphics:../Images/tricks_gr_299.gif]

the matrix becomes

[Graphics:../Images/tricks_gr_300.gif]
[Graphics:../Images/tricks_gr_301.gif]

and the perturbed eigenvalues are

[Graphics:../Images/tricks_gr_302.gif]
[Graphics:../Images/tricks_gr_303.gif]

Defining

[Graphics:../Images/tricks_gr_304.gif]

we can estimate the perturbed values from

[Graphics:../Images/tricks_gr_305.gif].

[Graphics:../Images/tricks_gr_306.gif]
[Graphics:../Images/tricks_gr_307.gif]

The results are in good agreement.

[Graphics:../Images/tricks_gr_308.gif]
[Graphics:../Images/tricks_gr_309.gif]

Document converted by Mathematica of Wolfram Research