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Mathematical Methods of Physics: For Undergraduate

Sunil Karna

Exercises 2.8 Exercise

1.

Find the eigen values and eigen vectors of the given matrix
\begin{equation*} \begin{bmatrix} 1 & -1 & 1\\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{bmatrix}. \end{equation*}
Answer.
Here, \(\lambda = 0,1,2;\) for
\begin{equation*} \lambda_{1}=0, X_{1}=\begin{bmatrix} 0\\1\\1 \end{bmatrix}, \end{equation*}
for
\begin{equation*} \lambda_{2}=1, \hspace{0.2cm} X_{2}=\begin{bmatrix} 1\\-1\\-1 \end{bmatrix}, \end{equation*}
for
\begin{equation*} \lambda_{3}=2,\hspace{0.2cm} X_{3}=\begin{bmatrix} 1\\-1\\0 \end{bmatrix}. \end{equation*}

2.

Find \(x{,}y{,}z\) such that
\begin{equation*} \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & -4 \\-1 & 3 & - 2 \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix} = \begin{bmatrix} 1 \\ -3 \\6 \end{bmatrix} \end{equation*}
Answer.
\begin{equation*} X = \begin{bmatrix} -1\\3\\2 \end{bmatrix} \end{equation*}

3.

Find the matrix that diagonalize the matrix
\begin{equation*} A=\begin{bmatrix} 1 & i & 1 \\ -i & 0 & 0 \\1 & 0 & 0 \end{bmatrix}. \end{equation*}
Answer.
\begin{equation*} P=\begin{bmatrix} 0 & 1 & 2i \\ i & -i & 1 \\ 1 & 1 & i \end{bmatrix}.\\ \end{equation*}

4.

Show that the matrix
\begin{equation*} A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \end{equation*}
is orthogonal.

5.

Diagonalizing matrix of a real symmetric matrix is orthogonal.

6.

Verify that the adjoint of a diagonal matrix of order three is a diagonal matrix.

7.

Find the inverse of
\begin{equation*} A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\3 & 5 & 6 \end{bmatrix}. \end{equation*}
Answer.
\begin{equation*} A^{-1} = \begin{bmatrix} 1 & -3 & 2 \\ -3 & 3 & -1 \\ 2 & -1 & 0 \end{bmatrix} \end{equation*}

8.

If
\begin{equation*} A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{bmatrix} \end{equation*}
and
\begin{equation*} B = \begin{bmatrix} 2 & 5 & 3 \\ 3 & 1 & 2 \\ 1 & 2 & 1 \end{bmatrix} \end{equation*}
then show that \((AB)^{-1} = B^{-1}A^{-1}\text{.}\)

9.

Show that the eigen values of a unitary matrix have unit modulus.

10.

Show that the eigen values of an orthogonal matrix have unit modulus.

11.

Show that a product of unitary matrices is unitary.
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