Matrices

Column and Row vector

Here are our definitions of a column vector and a row vector:

$$\text{column vector} = |y\rangle = \begin{bmatrix} a_1 \\ a_2 \\ \vdots\\ a_n \end{bmatrix}$$

$$\text{row vector} = \langle x| = \begin{bmatrix} a_1,a_2,\cdots,a_n \end{bmatrix}.$$

The vector and row vectors can be represented by one-dimenstional matrices.

Multiplying Vectors

A bracket $\langle x|y\rangle$, is essentially matrix multiplication of a row vector and a column vector. Here is our definition

$$\langle x|y\rangle = \begin{bmatrix} x_1,x_2,\cdots,x_n \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \\ \vdots\\ y_n \end{bmatrix} =x_{1}\cdot y_{1}+x_{2}\cdot y_{2}+\cdots x_{n}\cdot y_{n}$$

for example,

$$\langle y|x\rangle = \begin{bmatrix} 3 \ 2 \ 1 \ 4 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} =16$$

Matrix-vector multiplication

If the matrix and column vector are variables, we can write out the product this way: $$ A|x\rangle $$ This denotes the matrix $A$ multiplying the vector $|x\rangle$.

Let's say we have

$$A|x\rangle = \begin{bmatrix} 4 & 3 & 2\\ 4 & 1 & 3\\ 2 & 4 & 2 \end{bmatrix} \cdot \begin{bmatrix} w\\ y\\ z \end{bmatrix} = \begin{bmatrix} a\\ b\\ c \end{bmatrix}, \quad x = \begin{bmatrix} w\\ y\\ z \end{bmatrix}$$

we can seperate matrix $A$ into row vectors

$$\begin{array}{c} \begin{bmatrix} 4 & 3 & 2 \end{bmatrix} = \langle R_{1}|, \quad \begin{bmatrix} 4 & 3 & 3 \end{bmatrix} = \langle R_{2}|, \quad \begin{bmatrix} 2 & 4 & 2 \end{bmatrix} = \langle R_{3}|. \quad \end{array}$$

and we can put things together as

$$A|x\rangle = \begin{bmatrix} 4 & 3 & 2\\ 4 & 1 & 3\\ 2 & 4 & 2 \end{bmatrix} \cdot \begin{bmatrix} w\\ y\\ z \end{bmatrix} = \begin{bmatrix} \langle R_{1}|x\rangle \\ \langle R_{2}|x\rangle\\ \langle R_{3}|x\rangle \end{bmatrix}$$

In the same fashion, if we perform the product on the two following matrices $A$ and $B$ such that

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \quad B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6. \end{bmatrix}$$

Same, we rewrite matirx $A$ into row vectors $\langle R_{1}|,\langle R_{2},\langle R_{3}|$ and $B$ into column vectors $|C_{1}\rangle, |C_{2}\rangle, |C_{3}\rangle$. We can have the product of $A$ and $B$ matrix as

$$A \cdot B = \begin{bmatrix} \langle R_{1}|C_{1}\rangle & \langle R_{1}|C_{2}\rangle & \langle R_{1}|C_{3}\rangle \\ \langle R_{2}|C_{1}\rangle & \langle R_{2}|C_{2}\rangle & \langle R_{2}|C_{3}\rangle \end{bmatrix}$$

Quantum example

in quantum circuits, the inputs are qubits (vectors), and the gates are matrices. An example quantum logic gate, NOT gate, is show as

The NOT gate in quantum copmuting is represented by the following matrix:

$$X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$

if our input qubits is a $|1\rangle$, then the output would be:

$$X|1\rangle = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = |0\rangle$$

Hilbert space quantum

Hermitian matrix

A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose - that is, the element in the $i$-th row and $j$-th column is equal to the complex conjugate of the element in the $j$-th row and $i$-th column, for all indices $i$ and $j$. A is a Hermitian matrix $A$ contains elements $a_{ij}$, then

$$a_{ij} = \overline{a_{ij}}$$

or in a quantum mechanics notation

$$A^{H} = \overline{A^{T}} = A^{\dagger} .$$

For example, we have a complex square matrix $A$ such that

$$A = \begin{bmatrix} 0 & a-ib & c-id \\ a+ib & 1 & m-in \\ c+id & m+in & 2 \end{bmatrix}$$

we apply transpose, it becomes

$${A^{T}} = \begin{bmatrix} 0 & a+ib & c+id \\ a-ib & 1 & m+in \\ c-id & m-in & 2 \end{bmatrix}$$

then we find its cimplex conjugate, we have

$$\overline{A^{T}} = \begin{bmatrix} 0 & a-ib & c-id \\ a+ib & 1 & m-in \\ c+id & m+in & 2 \end{bmatrix} = A^{\dagger} = A$$

Diagonal values

The entries on the main diagonal of any Hermitian matrix are real.

Symmetric

A matrix that has only real entries is symmetric if and only if it is a Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix.

Normal

Every Hermitian matrix is a normal matrix. $AA^{\dagger} = A^{\dagger}A$.

Properties

For any Hermitian matrix $A$ and $B$, we have

1. $(A+B)_{ij} = A_{ij}+B_{ij} = \overline{A_{ij}}+\overline{B_{ij}} = \overline{A+B}_{ij}$
2. For an invertible Hermitian matrix. $A^{-1} = (A^{-1})^{H}$

Normal matrix

A complex square matrix $A$ is normalif it commutes with its conjugate transpose $A^{*}$.

$$A^{*}A = AA^{*}$$

Conjugate transpose

To maintain the consistancy, symbol $H$ has been replaced to $\dagger$.

A square matrix $A$ with entries $a_{ij}$ is called

1. Hermitian or self-adjoint if $A = A^{\dagger}$; i.e., $a_{ij} = \overline{a_{ji}}$.
2. Skew Hermitian if $A = -A^{\dagger}$; i.e., $a_{ij} = -\overline{a_{ji}}$.
3. Normal if $A^{\dagger}A = AA^{\dagger}$.
4. Unitary if $A^{\dagger} = A^{-1}$; i.e., $AA^{\dagger} = A^{\dagger}A = I$.

Unitary matrix

In linear algebra, an invertible complex square matrix $U$ is unitary if its inverse matrix $U^{-1}$ equals its conjugate transpose $U^{*}$, that is, if

$$U^{*}U = UU^{*} = I$$

In quantum, we use $\dagger$, that is

$$U^{\dagger}U = UU^{\dagger} = I$$

For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability.

Orthogonal matrix

In liner algrbra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

Also, it holds that

$$Q^{T}Q = QQ^{T} = I$$

where $Q^{T}$ is the transpose of $Q$ and $I$ is the identity matrix. As a result, it also holds that

$$Q^{T} = Q^{-1}.$$

Orthonormal basis

An orthonormal basis is a set of vectors have a norm of 1 and are pairwise orthogonal. ref

References:

1. Woody III, L. S. (2021). Essential mathematics for quantum computing. Packt Publishing. https://www.packtpub.com/en-us/product/essential-mathematics-for-quantum-computing-9781801070188
2. Hidary, J. D. (2019). Quantum computing: An applied approach. Springer. https://link.springer.com/book/10.1007/978-3-030-23922-0
3. Conjugate transpose (Wiki)
4. Orthogonal matrix (Wiki)
5. Hermitian matrix (Wiki)
6. Orthonormal basis
7. Orthonormal basis
8. Hilbert Space Quantum Mechanics