91Ó°ÊÓ

Test the set of linear homogeneous equations $$x+3 y+3 z=0, \quad x-y+z=0, \quad 2 x+y+3 z=0$$ to see if it possesses a nontrivial solution and find one.

Given the pair of equations $$x+2 y=3, \quad 2 x+4 y=6,$$ (a) show that the determinant of the coefficients vanishes; (b) show that the numerator determinants (Eq. (3.18)) also vanish; (c) find at least two solutions.

Show that \((\mathrm{AB})^{\dagger}=\mathrm{B}^{\dagger} \mathrm{A}^{\dagger}\).

As a converse of the theorem that Hermitian matrices have real eigenvalues and that eigenvectors corresponding to distinct eigenvalues are orthogonal, show that if (a) the eigenvalues of a matrix are real, and (b) the eigenvectors satisfy \(\mathbf{r}_{i}^{\dagger} \mathbf{r}_{j}=\delta_{i j}=\left\langle\mathbf{r}_{i} \mid \mathbf{r}_{j}\right\rangle\), then the matrix is Hermitian.

If \(A\) is orthogonal and \(\operatorname{det} A=+1\), show that \((\operatorname{det} A) a_{i j}=C_{i j}\), where \(C_{i j}\) is the cofactor of \(a_{i j}\). This yields the identities of Eq. (1.46) used in Section \(1.4\) to show that a cross product of vectors (in three-space) is itself a vector.

If \(A\) is a \(2 \times 2\) matrix show that its eigenvalues \(\lambda\) satisfy the equation $$ \lambda^{2}-\lambda \operatorname{trace}(A)+\operatorname{det} A=0 $$

A matrix \(C=S^{\dagger} S\). Show that the trace is positive definite unless \(S\) is the null matrix, in which case trace \((C)=0\).

Show that a real matrix that is not symmetric cannot be diagonalized by an orthogonal similarity transformation.

Another set of Euler rotations in common use is: (1) a rotation about the \(x_{3}\) -axis through an angle \(\varphi\), counterclockwise; (2) a rotation about the \(x_{1}^{\prime}\) -axis through an angle \(\theta\), counterclockwise; (3) a rotation about the \(x_{3}^{i j}\) -axis through an angle \(\psi\), counterclockwise. If $$ \begin{array}{ll} \alpha=\varphi-\pi / 2 & \varphi=\alpha+\pi / 2 \\ \beta=\theta & \theta=\beta \\ \gamma=\psi+\pi / 2 & \psi=\gamma-\pi / 2, \end{array} $$ show that the final systems are identical.

(a) Complex numbers, \(a+i b\), with \(a\) and \(b\) real, may be represented by (or are isomorphic with) \(2 \times 2\) matrices: $$ a+i b \leftrightarrow\left(\begin{array}{rr} a & b \\ -b & a \end{array}\right) . $$ Show that this matrix representation is valid for (i) addition and (ii) multiplication. (b) Find the matrix corresponding to \((a+i b)^{-1}\).