Questions 11 - 16#

Q11#

Find ABAB and BABA if AA=[1234] and BB=[5678] . Do these matrices commute and if not what is [ABAB]?

Q12#

(a) Find AA2,BB2,ABAB,BABA,AA2BB,and BB2AA if AA=[1101] and BB=[1011].

(b) Do AA2BB and BB2AA commute ?

Q13#

If xx=[x1,x2],yy=[y1,y2]T ,and QQ=[1496], find xQyxQy and yQxyQx.

Q14#

(a) Explain why if j is a constant factor that |jMM|=jn|MM| for an n×n matrix.

(b) Confirm equation 4 if AA=[abcd] by calculating AA1 and |AA1|.

(c) Using Sympy (or by hand ) find AA1 and |AA1| for the 3×3 matrix AA=[0bcb0dcd0].

Strategy: See Section 4.14, but now the problem is algebraic not numerical.

Q15 Commuting operators#

(a) Show that if PP and QQ are linear operators, not necessarily matrices, [PP,QQ]=[QQ,PP].

(b) (i) find [d/dx,x]sin(x) , (ii) [d/dx,x]f(x), and (iii) [d/dx,x] and

(c) Show that operators d/dx and x do not commute.

(d) Write a Python/Sympy commutator function and test your results.

(e) Do df(x)dx and abf(x)dx commute?

(f) Does the operator df/dx and a displacement operator Δf=f(x+c) commute?

(g) Does the operator xx which means multiply twice by x, commute with the inversion operator, Inv(f(x))=f(x)?

Q16 Commuting matrices#

BB and CC are commuting square matrices and the matrix AA is defined as AAeBB=11+BB+BB2/2!+ show that:

(a) eBBeCC=eB+CB+C,

(b) AA1=eBB,

(c) eCBCCBC1=CACCAC1.

Strategy: (a) Expand out the exponentials as shown in Section 5.5 and collect terms and try to reform an exponential series in B+CB+C. (b) Use the fact that for an inverse matrix AAAA1=11. (c) The expression CACCAC1 is a similarity transform and is itself a square matrix. To prove this result, expand the exponential on both sides of the equation