Calculate the following quantities for the matrices:
$$A\doteq\pmatrix{1&0&0\cr 0&0&1\cr 0&-1&0\cr}\qquad\qquad B\doteq\pmatrix{a&b&c\cr d&e&f\cr g&h&j\cr}\qquad\qquad C\doteq\pmatrix{\cos\theta&-\sin\theta\cr \sin\theta&\cos\theta\cr}$$
and the vectors:
$$\left|D\right\rangle\doteq\pmatrix{1\cr i\cr -1\cr}\qquad\qquad \left|E\right\rangle\doteq\pmatrix{1\cr i\cr}\qquad\qquad \left|F\right\rangle\doteq\pmatrix{1\cr -1\cr}$$
$AB$
${\rm tr} (AB)$
$A^{\dagger}$
$C^{-1}$
$A\vert D\rangle$
$\vert E\rangle^{\dagger}\equiv\langle E\vert$
$\langle D\vert A\vert D\rangle$
$\det(\lambda{\cal I}-A)$ where $\lambda$ is a scalar.
$\left(A\vert D\rangle\right)^{\dagger}$
Using explicit matrix multiplication (without using a theorem) verify that $\left(A\vert D\rangle\right)^{\dagger}=\langle D\vert A^{\dagger}$
The Pauli spin matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$ are defined by:
$$\sigma_x=\pmatrix{0&1\cr 1&0\cr}\qquad\qquad \sigma_y=\pmatrix{0&-i\cr i&0\cr}\qquad\qquad \sigma_z=\pmatrix{1&0\cr 0&-1\cr}$$
These matrices are related to angular momentum in quantum mechanics. Prove, and become familiar with, the identities listed below.
Show that each of the Pauli matrices is hermitian. (A matrix is hermitian if it is equal to its hermitian adjoint.
Show that the determinant of each of the Pauli matrices is $-1$.
Show that $\sigma_i^2={\cal I}$ for each of the Pauli matrices, i.e.\ for $i\in\left\{x,y,z\right\}$.
The Pauli spin matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$ are defined by:
$$\sigma_x=\pmatrix{0&1\cr 1&0\cr}\qquad\qquad \sigma_y=\pmatrix{0&-i\cr i&0\cr}\qquad\qquad \sigma_z=\pmatrix{1&0\cr 0&-1\cr}$$
These matrices are related to angular momentum in quantum mechanics. Prove, and become familiar with, the identities listed below.
Show that $\sigma_x \sigma_y = i\sigma_z$ and $\sigma_y \sigma_x = -i\sigma_z$. (Note: These identities also hold under a cyclic permutation of $\left\{x,y,z\right\}$, e.g.\ $x\rightarrow y$, $y\rightarrow z$, and $z\rightarrow x$).
The commutator of two matrices $A$ and $B$ is defined by $\left[A, B\right]\buildrel \rm def \over = AB-BA$. Show that $\left[\sigma_x, \sigma_y\right] = 2i\sigma_z$. (Note: This identity also holds under a cyclic permutation of $\left\{x,y,z\right\}$, e.g.\ $x\rightarrow y$, $y\rightarrow z$, and $z\rightarrow x$).
The anti-commutator of two matrices $A$ and $B$ is defined by $\left\{A, B\right\}\buildrel \rm def \over = AB+BA$. Show that $\left\{\sigma_x, \sigma_y\right\} = 0$. (Note: This identity also holds under a cyclic permutation of $\left\{x,y,z\right\}$, e.g.\ $x\rightarrow y$, $y\rightarrow z$, and $z\rightarrow x$).
Perform the following matrix multiplications:
$$\frac{1}{\sqrt{2}}\pmatrix{1&1\cr1&-1}\quad \pmatrix{3&1\cr1&3}\quad \frac{1}{\sqrt{2}}\pmatrix{1&1\cr1&-1}$$