resolvent matrix

The resolvent matrix of a matrix A is defined as


Note: I is the identity matrixMathworldPlanetmath and s is a complex variable. Also note that RA(s) is undefined on Sp(A) (the spectrum of A).

More generally, let A be a unital algebra over the field of complex numbersMathworldPlanetmathPlanetmath . The resolvent Rx of an element xA is a function from -Sp(x) to A given by


where Sp(x) is the spectrum of x: Sp(x)={tt1-x is not invertible in A}.

If A is commutativePlanetmathPlanetmathPlanetmath and sSp(x)Sp(y), then Rx(s)-Ry(s)=Rx(s)Ry(s)(x-y).

Title resolvent matrix
Canonical name ResolventMatrix
Date of creation 2013-03-22 13:36:20
Last modified on 2013-03-22 13:36:20
Owner mps (409)
Last modified by mps (409)
Numerical id 8
Author mps (409)
Entry type Definition
Classification msc 47A10
Classification msc 15A15
Defines resolvent