Questions 11 - 16

Q11

Find \(\pmb{AB}\) and \(\pmb{BA}\) if \(\displaystyle \pmb{A} =\begin{bmatrix} 1 & 2\\ 3& 4\end{bmatrix}\) and \(\displaystyle \pmb{B} =\begin{bmatrix} 5 & 6\\ 7& 8\end{bmatrix}\) . Do these matrices commute and if not what is \([\pmb{AB}]\)?

Q12

(a) Find \(\pmb{A}^2,\pmb{B}^2,\pmb{AB},\pmb{BA},\pmb{A}^2\pmb{B}\),and \(\pmb{B}^2\pmb{A}\) if \(\displaystyle \pmb{A} =\begin{bmatrix} 1 & 1\\ 0& 1\end{bmatrix}\) and \(\displaystyle \pmb{B} =\begin{bmatrix} 1 & 0\\ 1& 1\end{bmatrix}\).

(b) Do \(\pmb{A}^2\pmb{B}\) and \(\pmb{B}^2\pmb{A}\) commute ?

Q13

If \(\pmb{x}=[x_1, x_2 ],\pmb{y}=[y_1,y_2]^T\) ,and \(\displaystyle \pmb{Q}=\begin{bmatrix} 1 &-4\\ -9& 6\end{bmatrix}\), find \(\pmb{xQy}\) and \(\pmb{yQx}\).

Q14

(a) Explain why if \(j\) is a constant factor that \(|j\pmb{M}| = j^n|\pmb{M}|\) for an \(n \times n\) matrix.

(b) Confirm equation 4 if \(\displaystyle \pmb{A} =\begin{bmatrix} a & b\\ c& d\end{bmatrix}\) by calculating \(\pmb{A}^{-1}\) and \(| \pmb{A}^{-1} |\).

(c) Using Sympy (or by hand ) find \(\pmb{A}^{-1}\) and \(|\pmb{A}^{-1} |\) for the \(3\times 3\) matrix \(\displaystyle \pmb{A} =\begin{bmatrix} 0 & b & c\\ b& 0 & d\\ c & d & 0\end{bmatrix}\).

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

Q15 Commuting operators

(a) Show that if \(\pmb{P}\) and \(\pmb{Q}\) are linear operators, not necessarily matrices, \([\pmb{P},\pmb{Q}]=- [\pmb{Q},\pmb{P}]\).

(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 \(\displaystyle \frac{df(x)}{dx}\) and \(\displaystyle \int_a^bf(x)dx\) commute?

(f) Does the operator \(df/dx\) and a displacement operator \(\Delta f = f (x + c)\) 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

\(\pmb{B}\) and \(\pmb{C}\) are commuting square matrices and the matrix \(\pmb{A}\) is defined as \(\pmb{A} \equiv e^{\pmb{B}} =\pmb{1}+\pmb{B}+\pmb{B}^2/2! +\cdots \) show that:

(a) \(e^{\pmb{B}}e^{\pmb{C}} = e^{\pmb{B+C}}\),

(b) \(\pmb{A}^{-1} = e^{-\pmb{B}}\),

(c) \(e^{\pmb{CBC}^{-1}} = \pmb{CAC}^{-1}\).

Strategy: (a) Expand out the exponentials as shown in Section 5.5 and collect terms and try to reform an exponential series in \(\pmb{B + C}\). (b) Use the fact that for an inverse matrix \(\pmb{AA}^{-1} = \pmb{1}\). (c) The expression \(\pmb{CAC}^{-1}\) is a similarity transform and is itself a square matrix. To prove this result, expand the exponential on both sides of the equation