Question

5. Let V be quadratic polynomials on the interval [-1,1], with the inner product 〈f,g):= | f(t)g(t)dt, and D VVfHf be the differentiation operator. (a) Find the Hermitian transpose (adjoint) D, which is determined by its action on a basis, by calculating D'(1), D*(x), D'(x2), explicitly. Find the eigenvalues and corresponding eigenfunctions of D* (c) Find (D)

Answer 1

by definition in the given problem <f,g>==(integral(-1 to 1)f(t).g(t))

let f(t)=at²+bt+c,g(t)=t

integral(-1,1){(at²+bt+c).(t)}

integral(-,1,1){(at³+bt²+ct)}

at⁴/4+bt³/3+ct²/2===(1) is a Fourth order polynomial

(a)adj(D*)=co factor matrix of( D)

D=at⁴/4+bt³/3+ct²/2

D*=(D)^*=at⁴/4-bt³/3+ct²/2

in matrix form the equation itself is corresponding eigen values

D*(1)=a/4-b/3+c/2

D^*(x)=a/4x⁴-b/3x²+cx/2

D*(x²)=a/4x⁶-b/3x⁴/2+cx²/2

Figen values are found by the equation Dx-lamda I(x)=0 is a fourth order equation

D(x)=Lamda I(x)

D=Lamda Eigen Values

(a) D*(1)== Lamda values are {a/4,-b/3,c/2}

(c){D*(x)}³={(a/4)³,(-b/3)³,(c/2)³}