write out all the possibilities for chain structures for an

write out all the possibilities for chain structures for an eigenvalue of multiplicity.

Solution

For (A) = {1, 2, . . . , s} , we adopt the following definitions.

• The algebraic multiplicity of is the number of times it is repeated as a root of the characteristic polynomial. In other words, alg multA (i) = ai if and only if (x 1) a1 · · ·(x s) as = 0 is the characteristic equation for A.

• When alg multA () = 1, is called a simple eigenvalue.

• The geometric multiplicity of is dim N (A I). In other words, geo multA () is the maximal number of linearly independent eigenvectors associated with .

• Eigenvalues such that alg multA () = geo multA () are called semisimple eigenvalues of A. It follows from (7.2.2) on p. 511 that a simple eigenvalue is always semisimple, but not conversely.