Express f(A)as a linear combination of the matrices {I, A, A^2 }and write
f(A) = R(A) = sin At =c 0 I+c 1 A+c 2 A^2 ,where c 0 , c 1 and c 2 are functions of tto be determined. Since there is a repeated root,
use differentiation and write the equations
R(x) = c 0 +c 1 x+c 2 x^2
R′(x) = c 1 + 2c 2 xto obtain the system of equations
R(λ 1 ) = f(λ 1 ) = sin t=c 0 +c 1 +c 2
R′(λ 1 ) = f′(λ 1 ) = cos t=c 1 + 2 c 2R(λ 3 ) = f(λ 3 ) = −sin t=c 0 −c 1 +c 2(10 .49)These are three independent equations which can be used to solve for the coefficients
c 0 , c 1 and c 2 .Solving these equations for c 0 , c 1 and c 2 one finds
c 0 =^1
2(sin t−cos t), c 1 = sin t, c 2 =^1
2(cos t−sin t)and consequently the matrix function sin At has the representation
sin At =(
sin t−cos t
2)
I+Asin t+1
2 (cos t−sin t)A(^2).
An alternate form for this result is the matrix form
sin At =^1
2
0 2 sin t cos t−sin t
cos t+ sin t cos t−sin t cos t+ sin t
2 cos t 2 cos t cos t+ sin t
A=
0 1 0
1 0 1
1 1 1