Square Matrices

Identity Matrix

The identity matrix has ones along the main diagonal and zeros everywhere else: \begin{equation} \mat{I} = \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix} \label{eq:identity} \end{equation}

Inverse

If a square matrix \(\mat{A}\) has an inverse \(\mat{A}^{-1}\), then \begin{equation} \mat{A}^{-1} \vdot \mat{A} = \mat{I} = \mat{A} \vdot \mat{A}^{-1} \end{equation}

If \(\mat{A}\) transforms a column vector \(\vb{x}\) to a new column vector \(\vb{y} = \mat{A} \vdot \vb{x}\), the inverse matrix allows us to recover \(\vb{x}\) from \(\vb{y}\) via \(\vb{x} = \mat{A}^{-1} \vdot \vb{y}\) since \begin{equation} \mat{A}^{-1} \vdot \vb{y} = \mat{A}^{-1} \vdot (\mat{A} \vdot \vb{x}) = (\mat{A}^{-1} \vdot \mat{A}) \vdot \vb{x} = \mat{I} \vdot \vb{x} = \vb{x} \end{equation} The existence of an inverse implies that operating with \(\mat{A}\) on a vector does not entail the loss of information; that is, the matrix has a trivial nullspace, which means that its rows (and columns) are linearly independent. It usually isn’t obvious by inspection whether a square matrix has an inverse (that its rows are linearly independent). If one or more rows can be expressed as the linear combination of other rows, then the matrix is singular, its determinant vanishes, and it does not have an inverse.

You can use Gauss-Jordan elimination to compute the inverse of a square matrix (provided it has one).

Hermitian Matrices

A Hermitian matrix is a complex matrix equal to its conjugate transpose: \(\mat{H} = \widetilde{\mat{H}} = (\mat{H}^{*})^{\mathrm{T}}\).

In matrix mechanics, Hermitian matrices represent physically observable quantities (e.g., angular momentum, the hamiltonian (energy), etc.). The matrix representing the \(z\) component of spin angular momentum of a spin-1/2 particle, in the basis of \(\ket{\uparrow}, \ket{\downarrow}\) along \(z\) is \[ \hat{S}_z \longrightarrow \frac{\hslash}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \] Note that the “hat” on \(S_z\) indicates that it is an operator; it operates on a vector in the space of the spin-1/2 particle and produces another vector in the space.
Hermitian matrices are diagonalizable.
Hermitian matrices have real eigenvalues.

Normal Matrices

A normal matrix commutes with its conjugate transpose: \(\widetilde{\mat{A}} \vdot \mat{A} = \mat{A} \vdot \widetilde{\mat{A}}\).

Unitary Matrices

A unitary matrix is a complex square matrix whose inverse is equal to its conjugate transpose: \(\mat{U}^{-1} = \widetilde{\mat{U}}\).

A unitary matrix that represents the rotation of a spin-1/2 particle through angle \(\phi\) around the \(z\) axis, expressed in the basis of \(\ket{\uparrow}, \ket{\downarrow}\) along \(z\) is \[ \hat{R}(\phi \vu{z}) \longrightarrow \begin{pmatrix} e^{-i\phi/2} & 0 \\ 0 & e^{i\phi/2} \end{pmatrix} \]
Unitary matrices are one type of normal matrices. They preserve lengths and inner products.
\(\mat{U} = e^{i \mat{H}}\), where \(\mat{H}\) is a Hermitian matrix. The meaning of the exponential of an operator derives from the exponential series: \begin{equation} e^{i\mat{H}} = \mat{I} + i\mat{H} + \frac{i^2}{2!} \mat{H}\vdot \mat{H} + \frac{i^3}{3!} \mat{H}\vdot\mat{H}\vdot\mat{H} + \cdots \end{equation}

Orthogonal Matrices

An orthogonal matrix satisfies \begin{equation} \mat{O}^{\rm T} \vdot \mat{O} = \mat{O} \vdot \mat{O}^{\rm T} = \mat{I} \end{equation} where \(\mat{O}^{\rm T}\) is the transpose of \(\mat{O}\), which interchanges rows and columns. For an orthogonal matrix, therefore, \begin{equation} \mat{O}^{\rm T} = \mat{O}^{-1} \end{equation}

The determinant of an orthogonal matrix is either \(+1\) or \(-1\). Orthogonal matrices with determinant \(+1\) are called special orthogonal matrices [the group of all such matrices of dimension \(n\) is notated \(\mat{SO}(n)\)] and they correspond to rotations in \(\mathbb{R}^n\). Orthogonal matrices with determinant equal to \(-1\) combine rotations with an inversion through the origin. They reverse the handedness of the underlying coordinate system.

Orthogonal matrices:

preserve lengths of vectors
preserve the inner product between two vectors

Rotation Matrices

The matrix \[ \mat{R}(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \] rotates a column vector through angle \(\theta\) in the counterclockwise direction. You can confirm that \(\mat{R}(\pi/2)\) rotates \(\vu{x} \to \begin{pmatrix}1 \\\ 0\end{pmatrix}\) into \(\vu{y} \to \begin{pmatrix} 0 \\\ 1 \end{pmatrix}\) and \(\vu{y}\) into \(-\vu{x}\).

We can generalize readily to 3 dimensions, at least for rotations around one of the basis vectors. For example, \[ \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix} \] rotates a column vector around the \(y\) axis. All proper rotations (that don’t alter the handedness of the basis vectors) have a determinant of 1. Improper rotations, which do change the handedness of the basis vectors, have determinant \(-1\).

Determinants

You undoubtedly already know what a determinant is, and how you can work it out using minors, but allow me to make a couple of definitions.

Each contribution to the determinant of an \(N\)-dimensional matrix is the product of \(N\) terms, one from each column and row.
There are \(N!\) such contributions, since there are \(N\) choices for the column in the first row, \(N-1\) choices for the column in the second row, etc.
We can represent a particular ordering with the indices of the successive columns. For example, 1324 would mean to multiply \(a_{11} a_{23} a_{32} a_{44}\) for a matrix whose elements are \(a_{ij}\).
Starting from the ordering \(123\cdots\) for the columns, every other ordering can be constructed by pairwise exchange of two elements. For example, to get to 1423 from the reference ordering 1234, we could proceed \(1234 \to 1324 \to 1423\), which took two exchanges. All possible permutations take either an even or odd number of exchanges.
The determinant is the sum of all \(N!\) N-term products, which are counted positive for even permutations and negative for odd permutations.
The Levi-Civita symbol \(\varepsilon_{ijklm\cdots}\) for \(N\) elements has \(N\) subscripts (indices) and is defined by \begin{equation} \varepsilon_{ijk\cdots} = \begin{cases} 1 & ijk\cdots = 123\cdots \text{ and even permutations} \\ -1 & ijk\cdots = 132\cdots \text{ and odd permutations of } 123\cdots \\ 0 & \text{if any index is repeated} \end{cases} \end{equation}
The determinant is then defined by \begin{equation}\label{eq:determinant} D = \sum_{ijk\cdots} (\varepsilon_{ijk\cdots}) a_{1i} a_{2j} a_{3k} \cdots \end{equation}

Properties of Determinants

Exchanging two rows in a matrix changes the sign of the value of its determinant.
Exchanging two columns in a matrix changes the sign of the value of its determinant.
Multiplying one row of a matrix by a constant \(\alpha\) multiplies the determinant by \(\alpha\).
Multiplying on row of a matrix by a constant \(\alpha\) and adding the result to another row leaves the determinant unchanged.
Multiplying a matrix by \(\alpha\) multiplies the determinant by \(\alpha^N\): \(\det(\alpha \mat{A}) = \alpha^n \det(\mat{A})\).
The determinant of a product is the product of determinants: \(\det(\mat{A} \mat{B}) = \det(\mat{A}) \det(\mat{B})\).
The determinant of the inverse is the reciprocal of the determinant: \(\det(\mat{A}^{-1}) = \frac{1}{\det{\mat{A}}}\).

Trace

The trace of a square matrix is the sum of the elements on the main diagonal: \[ \text{Tr } \mat{A} = \sum_{j=1}^N a_{jj} \] The trace has the property \[ \text{Tr }(\mat{A \cdot B}) = \text{Tr }(\mat{B \cdot A}) \]

Next: Gauss-Jordan Elimination

Physics 64 / Linear Algebra