Mathematical Physics 202 - Part B Problem Sheet 1

Author

Nigel Hussey

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Question 1 - Determinant of a 3x3 matrix

Consider matrix A: \(\begin{pmatrix} 1 & 2 \phantom{-} & 5 \\ -1 \phantom{-} & 0 \phantom{-}& 1 \\ 3 & 1 \phantom{-}& 2 \end{pmatrix}\)

  1. Show that its determinant \(|A|\) is independent of choice of row or column.

  2. Show that \(|A| = |A^{T}|\), i.e. that the determinant of \(A\) is equal to the determinant of its transpose.

Question 2 - Elimination of a 3x3 matrix

Solve: \[\begin{aligned} 2x_{1} + x_{2} + 3x_{3} &=& 4 \\ 2x_{1} - 2x_{2} - x_{3} &=& 1 \\ -2x_{1} + 4x_{2} + x_{3} &=& 1 \end{aligned}\] Using the augmented matrix method.

Question 3 - Elimination of a 4x4 matrix

Show by elimination that \[\left( \begin{array}{cccc:c} 2 & 1 & -1 \phantom{-}& 1 \phantom{-}& -2 \phantom{-}\\ 1 & -1 \phantom{-}& -1 \phantom{-}& 1 \phantom{-}& 1 \\ 1 & -4 \phantom{-}& -2 \phantom{-}& 2 \phantom{-} & 6 \\ 4 & 1 & -3 \phantom{-}& 3 \phantom{-} & -1 \phantom{-} \end{array} \right)\] has no solution.

Question 4 - Diagonalising a 3x3 matrix

Consider the matrix \(\begin{pmatrix} 2 & 0 & 0 \\ 1 & 2 \phantom{-} & 1 \\ 1 & 3 \phantom{-} & 2 \end{pmatrix}\)

Recall that \(\Lambda = S^{-1}AS\) where \(\Lambda\) is a diagonal matrix whose elements are eigenvalues of \(A\), \(S\) is the corresponding eigenvector matrix and \(S^{-1}\) is its inverse. Diagonalise \(A\) using the following steps:

  1. Let \(\Lambda = \begin{pmatrix} \lambda_{1} & 0 & 0 \\ 0 & \lambda_{2} & 0 \\ 0 & 0 & \lambda_{3} \end{pmatrix}\) and find \(\lambda_{1}, \lambda_{2}, \lambda_{3}\) using \(|A-\Lambda I| = 0\).

  2. Determine the corresponding eigenvectors to determine \(S = \begin{pmatrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{pmatrix}\) Hint: set \(y_{1} = y_{2} = y_{3} = 1\).

  3. Determine \(S^{-1}\) using the four steps given in the second lecture.

  4. Check that \(S^{-1}AS = \Lambda\).

Question 5 - Pauli matrices

The three Pauli matrices are: \[\hat{\sigma_{1}} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad \hat{\sigma_{2}} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \quad \hat{\sigma_{3}} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\]

  1. Show that all three Pauli matrices are Hermitian, involuntary and unitary.

  2. Show that \(|\sigma_{i}| = -1\) and Tr\((\sigma_{i}) = 0\).

  3. Find the eigenvalues and (normalised) eigenvectors for \(\sigma_{2}\) and \(\sigma_{3}\).

  4. Show that the commutation relation \([\sigma_{i},\sigma_{j}] = 2i\sigma_{k}\) where \(i,j,k = 1,2,3\) ; \(2,3,1\) and \(3,1,2\) (Hint: \([\hat{A},\hat{B}] = \hat{A}\hat{B}-\hat{B}\hat{A})\).

  5. Show that the anti-commutation relation \({\hat{A}, \hat{B}}\) = \(\hat{A}\hat{B} +\hat{B}\hat{A}\) is \(\{\sigma_{i},\sigma_{j}\} = \{\sigma_{j},\sigma_{k}\} =\{\sigma_{k},\sigma_{i}\} = 0\).

Question 6 - Fibonacci sequence

The Fibonacci sequence \([0,1,1,2,3,5,8,12,\dots]\) can be expressed by the simple formula \(f_{k+2} = f_{k+1} + f_{k}\).

Let \(u_{k} = \begin{bmatrix} f_{k+1} \\ f_{k} \end{bmatrix}\) and \(u_{k+1} = \begin{bmatrix} f_{k+2} \\ f_{k+1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} f_{k+1} \\ f_{k} \end{bmatrix}\) (check this relationship)

  1. Find the eigenvalues of \(\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}\).

  2. Find the corresponding eigenvectors \(x_{1}\) and \(x_{2}\).

  3. Writing \(u_{0} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = c_{1}x_{1} + c_{2}x_{2}\), find \(c_{1}\) and \(c_{2}\).

  4. From the above, find \(f_{100}\).