Estimating eigenvalues Gershgorins theorem

With the significant effort required to find the roots of 

For matrices that are not triangular, we cannot determine the eigenvalues by inspection, but we can obtain upper and lower bounds using Gershgorin s theorem. Let be an iVx W 

Matrix norm, spectral radius, and condition number 

We have defined the norm of a vector as a rule that assigns to every v e a scalar w 9 that represents the size of v and that satisfies w 0, where w = 0 if and only if V = 0. For each particular vector norm v, we can generate a corresponding ia/nx norm,  

As the maximum is attained when points in the direction of an eigenvector for the eigenvalue of largest modulus. 

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The famous Gershgorin theorem gives estimates of eigenvalues. The estimates of correspondent eigenvectors are not so well-known. In the chapter we use some estimates of eigenvectors of kinetic matrices. Here we formulate and prove these estimates for general matrices. Below A = (a,y) is a complex n x n matrix. Pi — (sums of nondiagonal elements in rows), Qi — (sums of... 

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