Pauli and Dirac matrices

the Hermitian matrices of size $2 \times 2$ with zero trace form a linear subspace $V_{H}^{0}$ of the real space $V_{H}$ of all Hermitian matrices of size $2 \times 2$;

the dimension of the space $V_{H}^{0}$ equals three, and as a basis one can take the so-called Pauli matrices:

$V_{H}^{0}$ is a Euclidean space with respect to the inner product

where the Pauli matrices $S_{1}, S_{2}, S_{3}$ form an orthonormal basis of this space.

Show that the $2 \times 2$ matrices

are linearly independent.

Show that any $2 \times 2$ complex matrix has a unique representation of the form

for some $a_{0}, a_{1}, a_{2}, a_{3} \in \mathbb {C}$.

Show that the matrix can be written as a linear combination of the Pauli spin matrices $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ and the identity matrix $I_{2}$.

Find the inverse map.

Calculate the determinant of $2 \times 2$ hermitian matrix $H$. Discuss.

Find the eigenvalues and normalized eigenvectors of the $2 \times 2$ matrix

Are the eigenvectors orthogonal to each other?

Find the eigenvalues and normalized eigenvectors of the $2 \times 2$ matrix

Consider the normalized vector in $\mathbb {C}^{3}$

Calculate the $2 \times 2$ matrix $U(\theta ,\phi ) = \overrightarrow {n} \cdot \overrightarrow {\sigma } \equiv n_{1}\sigma_{1} + n_{2}\sigma_{2} + n_{3}\sigma_{3}$, where $\sigma_{1}, \sigma_{2}, \sigma_{3}$ are the Pauli spin matrices. Is the matrix $U(\theta ,\phi )$ unitary? Find the trace and the determinant. Is the matrix $U(\theta ,\phi )$ hermitian? Find the eigenvalues and normalized eigenvectors of $U(\theta ,\phi )$.

Let $\epsilon \in \mathbb {R}$. Find the eigenvalues and eigenvectors of the $2 \times 2$ matrix

For $\epsilon =0$ we obtain the Pauli spin matrix $\sigma_{1}$, for $\epsilon =1$ we have the Hadamard matrix and for $\epsilon \rightarrow \infty$ we obtain the Pauli spin matrix $\sigma_{3}$.

Let $\epsilon \in \mathbb {R}$. Find the eigenvalues and eigenvectors of the $2 \times 2$ matrix

The matrices $A(\epsilon )$ and $B(\epsilon )$ are connected via the invertible transformation $\epsilon \rightarrow \sinh (\epsilon )$.

First show that they are normalized and orthonormal.

Assume that for the vector $\overrightarrow {v}_{1}$ is an eigenvector with the corresponding eigenvalue $+1$ and that the vector $\overrightarrow {v}_{2}$ is an eigenvector with the corresponding eigenvalue $-1$. Apply the spectral theorem to find the corresponding $2 \times 2$ matrix.

Since the vectors $\overrightarrow {v}_{1}$ and $\overrightarrow {v}_{2}$ form an orthonormal basis in $\mathbb {C}^{2}$ we can form an orthonormal basis in $\mathbb {C}^{4}$ via the Kronecker product

Assume that the eigenvalue for $\overrightarrow {w}_{11}$ is $+1$, for $\overrightarrow {w}_{12}$ $-1$, for $\overrightarrow {w}_{21}$ $-1$ and for $\overrightarrow {w}_{22}$ $+1$. Apply the spectral theorem to find the corresponding $4 \times 4$ matrix.

Calculate $\sigma_{12}$, $\sigma_{13}$, $\sigma_{14}$, $\sigma_{23}$, $\sigma_{24}$, $\sigma_{34}$.

Do the 16 matrices

form a basis in the Hilbert space $M_{4}(\mathbb {C})$? If so is the basis orthogonal?

Consider the Hilbert space $\mathcal{H}$ of the $2 \times 2$ matrices over the complex numbers. Show that the rescaled Pauli spin matrices $\mu_{j} = \frac{1}{\sqrt{2}}\sigma_{j}$, $j = 1, 2, 3$

plus the rescaled $2 \times 2$ identity matrix

form an orthonormal basis in the Hilbert space $\mathcal{H}$.

Find an orthonormal basis given by hermitian matrices in the Hilbert space $\mathcal{H}$ of $4 \times 4$ matrices over $\mathbb {C}$.

Given the standard basis $\left\{ \overrightarrow {e}_{1}, \overrightarrow {e}_{2}\right\}$ in $\mathbb {C}^{2}$. Is

a basis in $\mathbb {C}^{4}$?

Let $\overrightarrow {e}_{1}$, $\overrightarrow {e}_{2}$ be the standard basis in $\mathbb {C}^{2}$. The eight vectors

provides the standard basis in $\mathbb {R}^{8}$. Do the 8 vectors

form an orthonormal basis in $\mathbb {R}^{8}$?

Let the vectors $\overrightarrow {v}_{1}$, $\overrightarrow {v}_{2}$ form an orthonormal basis in $\mathbb {C}^{2}$. Then the four vectors

form an orthonormal basis in $\mathbb {C}^{4}$. Do the four vectors

form a basis in $\mathbb {C}^{4}$? Can the four vectors be written as a Kronecker product of two vectors in $\mathbb {C}^{2}$?

The three vectors

form an orthonormal basis in $\mathbb {R}^{3}$. Do the 9 vectors

form an orthonormal basis in $\mathbb {R}^{9}$? These vectors play a role for the $\pi$-mesons.

Assume that

Find the coefficients $t_{jk\ell }$ such that

Assume that

Find the coefficients $t_{jk\ell }$ such that

Show that any $2 \times 2$ matrix can be written as linear combination

Show that any $4 \times 4$ matrix can be written as linear combination

Show that any $2^{n} \times 2^{n}$ matrix can be written as linear combination

Consider the 256 sixteen times sixteen matrices

where $j_{0}, j_{1}, j_{2}, j_{3} \in \left\{ 0,1,2,3\right\}$. Let

Find $\psi^{T} T_{j_{0}j_{1}j_{2}j_{3}} \psi = t_{j_{0}j_{1}j_{2}j_{3}}$.

Let $\overrightarrow {u}$, $\overrightarrow {v}$ be normalized column vectors in $\mathbb {C}^{n}$ with $\overrightarrow {u}^{*}\overrightarrow {v} = 0$. Show that $\overrightarrow {u} \otimes \overrightarrow {v} - \overrightarrow {v} \otimes \overrightarrow {u}$ cannot be written as

where $\overrightarrow {a}, \overrightarrow {b} \in \mathbb {C}^{n}$. This problem plays a role for entanglement.

Let $\overrightarrow {u}, \overrightarrow {v} \in \mathbb {C}^{n}$ and linearly independent. Can one find $\overrightarrow {x}, \overrightarrow {y} \in \mathbb {C}^{n}$ such that

Let $A$, $B$ be nonzero $n \times n$ matrices over $\mathbb {C}$. Assume that $A$ and $B$ are linearly independent. Show that $A \otimes B - B \otimes A$ cannot be written as $A \otimes B - B \otimes A = X \otimes Y$, where $X, Y$ are $n \times n$ matrices over $\mathbb {C}$.

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices. Solve the nonlinear eigenvalue problem

where $\overrightarrow {u} \in \mathbb {C}^{2}$ and $\overrightarrow {u} \neq \overrightarrow {0}$.

Solve the nonlinear eigenvalue problem

where $\overrightarrow {v} \in \mathbb {C}^{4}$ and $\overrightarrow {v} \neq \overrightarrow {0}$. Compare the result to (i). Discuss.

Find the eigenvalues and normalized eigenvectors of the $4 \times 4$ matrix

Find the eigenvalues and normalized eigenvectors of the $6 \times 6$ matrix

Can the results from (i) be utilized here?

Find the eigenvalues of the $6 \times 6$ matrices