16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 4.3.3

Polynomials

Mathematics — Linear Algebra · lagrange-polynomials-diagonal-matrix

Problem

Let

F

be the field of real numbers,

A = \begin{bmatrix} 2 & 0 & 0 & 0 \\ \bullet & 2 & 0 & 0 \\ \bullet & \bullet & 3 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \qquad p = (x-2)(x-3)(x-1).

((a)) Show that

p(A) = 0

. ((b)) Let

P_1, P_2, P_3

be the Lagrange polynomials for

t_1 = 2

t_2 = 3

t_3 = 1

. Compute

E_i = P_i(A)

i = 1, 2, 3

. ((c)) Show that

E_1 + E_2 + E_3 = I

E_i E_j = 0

i \neq j

E_i^2 = E_i

. ((d)) Show that

A = 2E_1 + 3E_2 + E_3