Suppose a 6x6 matrix has two eigenvalues A what can we say a

Suppose a 6x6 matrix has two eigenvalues

(A) what can we say about their multiplicities? What is the range of possible total defect,

i.e. the sum of their defect numbers? Can we say anything about the sum of the dimension of the two eigenspaces? What about the dimension of the eigenspaces and total defect?

Solution

A. since the matrix is 6X6 so the characteristic equation for the eigen values will be a 6 degree equation . So the multiplicity of eigen value can vary from 1 to 5 .because a 6 degree equation can have upto 6 roots.

B.There can be 4 defects as the matrix is 6X6 and there is only 2 eigen values . that is two linearly independent values.

C.Dimension of the eigenspace must be two corresponding to each eigen values. And the total defects will be 4.