Matrix Problems

Problem 1. Given a matrix A = (aij)n×n, state the definition of its eigenvectors and eigenvalues.

Problem 2. Given a matrix A = (aij)n×n, suppose A has n eigenvectors

V1, V2, . . . , Vn

with eigenvalues λ1, . . . ,λn,

respectively. Suppose the matrix whose column vectors are V1, . . ., Vn is an invertible matrix.

Explain why the solutions to the corresponding system of first order linear ODEs

X′ = AX

is given by X = C1e

λ1tV1 + C2e λ2tV2 + · · · + CneλntVn,

where C1, C2, · · · , Cn are arbitrary constants.

(Answer to this problem was explained in detail in lecture today.)

1