Matrices

Definition

Properties of Matrices

1. Properties of Addition

$$\boxed{A_{m \times n} + A_{m \times n} = A_{m \times n}}$$

1. Commutative: $A + B = B + A$
2. Associative: $A + (B + C) = (A + B) + C$

Properties of the Zero Matrix ($\theta$)

1. $A + \theta = \theta + A = A$
2. $\theta - A = -A$
3. $A + (-A) = A - A = \theta$
4. $A\theta = \theta$ ; $\theta A = \theta$

Where:

• $\theta =$ zero (null) matrix.
• $-A =$ additive inverse of $A$.

2. Properties of Multiplication

$$\boxed{A_{m \times n} \times B_{p \times q} = C_{m \times q}}$$

Condition: $n = p$ (number of columns of $A$ = rows of $B$).

1. Left distributive: $A(B + C) = AB + AC$
2. Right distributive: $(A + B)C = AC + BC$
3. Associative: $A(BC) = (AB)C$
4. $AI = A$ ; $I =$ identity matrix.
5. Multiplication is not commutative: $AB \neq BA$
6. Multiplication by a scalar ($k$):

   $$k \cdot A_{m \times n} = [k \cdot a_{ij}]$$

   Example:

   $$k \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} ka & kb \\ kc & kd \end{bmatrix}$$

7. $k(A + B) = kA + kB$
8. $k_1(k_2A) = (k_1k_2)A$

3. Properties of Powers

1. Zero power:

   $$A^0 = I$$

   Where $I$ is the identity matrix.

2. Positive power ($n > 0$):

   $$A^n = \underbrace{AAA \cdots A}_{n \text{ factors}}$$

3. Product of powers with the same base:

   $$A^r A^s = A^{r+s}$$

4. Power of a power:

   $$(A^r)^s = A^{rs}$$

4. Polynomial Matrix

Given a polynomial:

$$P(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$$

If $A$ is a matrix, the matrix evaluation of the polynomial is defined as:

$$P(A) = a_0I + a_1A + a_2A^2 + \cdots + a_nA^n$$

Where:

• $I$ is the identity matrix of the same size as $A$.
• The terms $A^k$ are matrix powers.

Types of Matrices

1. Square Matrix

A matrix is square when the number of rows equals the number of columns.
Notation:

$$A_{n \times n} = A_n = [a_{ij}] \in \mathbb{R}^n$$

where:

$$1 \leq i \leq n \quad \text{and} \quad 1 \leq j \leq n$$

2. Zero (Null) Matrix ($\theta$)

Matrix where all its elements are zero:

$$A_{m \times n} = \theta = [a_{ij} = 0] = \begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}$$

3. Identity Matrix ($I_n$)

Square matrix with ones on the main diagonal and zeros elsewhere:

$$I_{n \times n} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Formal definition:

$$I_{n \times n} = \begin{cases} a_{ij} = 1 & \text{if } i = j \\ a_{ij} = 0 & \text{if } i \neq j \end{cases}$$

4. Row Matrix

Matrix with a single row:

$$A_{1 \times n} = [a_{1j}] \in \mathbb{R}^{1 \times n}$$

Explicit example:

$$A_{1 \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \end{bmatrix}$$

5. Column Matrix

Matrix with a single column:

$$A_{m \times 1} = [a_{i1}] \in \mathbb{R}^{m \times 1}$$

General form:

$$A_{m \times 1} = \begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix}$$

6. Transpose Matrix ($A^t$)

Given a matrix $A_{m \times n}$, its transpose $A^t$ is of size $n \times m$, where rows and columns are swapped:

Definition:

$$A = [a_{ij}] \implies A^t = [a_{ji}]$$

Example:
If $A = \begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix}$, then:

$$A^t = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}$$

7. Triangular (Echelon) Matrix

Special case: If $AB = \theta$ (zero matrix), it does not imply that $A$ or $B$ are zero.

Example:

$$\begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$

8. Upper Triangular Matrix

Square matrix where elements below the main diagonal are zero:

$$A_{n \times n} = \begin{cases} a_{ij} \neq 0 & \text{if } i \leq j \\ a_{ij} = 0 & \text{if } i > j \end{cases}$$

Example ($3 \times 3$):

$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{bmatrix}$$

9. Lower Triangular Matrix

Square matrix where elements above the main diagonal are zero:

$$A_{n \times n} = \begin{cases} a_{ij} \neq 0 & \text{if } i \geq j \\ a_{ij} = 0 & \text{if } i < j \end{cases}$$

Example ($3 \times 3$):

$$\begin{bmatrix} a_{11} & 0 & 0 \\ a_{21} & a_{22} & 0 \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$

10. Diagonal Matrix

A diagonal matrix is a square matrix that is simultaneously upper and lower triangular, where all elements outside the main diagonal are zero.

General definition:

$$D_{n \times n} = \begin{bmatrix} d_1 & 0 & \cdots & 0 \\ 0 & d_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_n \end{bmatrix}$$

Formal definition:

$$D_{n \times n} = \begin{cases} a_{ij} \neq 0 & \text{if } i = j \\ a_{ij} = 0 & \text{if } i \neq j \end{cases}$$

Power property:
The $k$-th power of a diagonal matrix is calculated by raising each diagonal element to that power:

$$D^k = \begin{bmatrix} d_1^k & 0 & \cdots & 0 \\ 0 & d_2^k & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_n^k \end{bmatrix}$$

Example:

$$D = \begin{bmatrix} -5 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 7 \end{bmatrix}$$

11. Inverse Diagonal Matrix

For a non-singular diagonal matrix (where all diagonal elements $d_i \neq 0$), its inverse is calculated by taking the reciprocal of each diagonal element:

$$D^{-1} = \begin{bmatrix} \frac{1}{d_1} & 0 & \cdots & 0 \\ 0 & \frac{1}{d_2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \frac{1}{d_n} \end{bmatrix}$$

Condition: All $d_i$ must be non-zero for $D^{-1}$ to exist.

12. Conjugate Matrix

Given a matrix $A$ with complex elements, its conjugate matrix $\overline{A}$ is obtained by conjugating each element of $A$.

Example:

$$A = \begin{bmatrix} i & 1 - i \\ 3 & 1 - 2i \end{bmatrix} \implies \overline{A} = \begin{bmatrix} -i & 1 + i \\ 3 & 1 + 2i \end{bmatrix}$$

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Special Matrices

1. Symmetric Matrix

A square matrix $A_{n \times n} = [a_{ij}]$ is symmetric if and only if:

$$A = A^t$$

2. Antisymmetric (Skew-Symmetric) Matrix

A square matrix $A_{n \times n} = [a_{ij}]$ is antisymmetric if and only if:

$$A^t = -A$$

3. Normal Matrix

A matrix is normal if it commutes with its transpose:

$$A \cdot A^t = A^t \cdot A$$

Includes:

• Symmetric matrices
• Antisymmetric matrices
• Unitary matrices

4. Singular Matrix

A square matrix is singular if:

$$\det(A) = 0$$

Consequence:

• It does not have an inverse
• Its rank is less than $n$

5. Regular (Non-singular) Matrix

A square matrix is regular if:

$$\det(A) \neq 0 \quad \text{and} \quad \rho(A) = n$$

6. Periodic Matrix

A matrix is periodic if there exists $k \in \mathbb{Z}^+$ such that:

$$A^{k+1} = A$$

Special case:
If $A^k = I$, it is said to have period $k$

7. Idempotent Matrix

A square matrix is idempotent if:

$$A^2 = A$$

8. Nilpotent Matrix

A square matrix $A$ is nilpotent if there exists an integer $k \geq 2$ such that:

$$A^k = \theta$$

where $\theta$ is the zero matrix. The smallest $k$ that satisfies this condition is called the index of nilpotency.

9. Involutory Matrix

A square matrix $A$ is involutory if it satisfies:

$$A^2 = I$$

That is, its square is the identity matrix.

Behavior for powers:

• $A^k = A$ if $k$ is odd
• $A^k = I$ if $k$ is even

10. Orthogonal Matrix

A square matrix $A$ is orthogonal if it satisfies:

$$A A^t = A^t A = I$$

which implies that:

$$A^{-1} = A^t$$

Fundamental relation:

$$\sin^2 x + \cos^2 x = 1$$

11. Hermitian Matrix

A complex square matrix $A \in \mathbb{C}^{n \times n}$ is Hermitian if it coincides with its conjugate transpose:

$$A = A^* \quad \text{where} \quad A^* = (\overline{A})^t$$

12. Skew-Hermitian (Antihermitian) Matrix

A complex square matrix $A \in \mathbb{C}^{n \times n}$ is skew-Hermitian if:

$$A = -A^* \quad \text{where} \quad A^* = (\overline{A})^t$$

Relation between Hermitian and Skew-Hermitian matrices:

• Any complex matrix $B$ can be written as:

  $$B = H + K$$

  where $H$ is Hermitian ($H = \frac{B + B^*}{2}$) and $K$ is skew-Hermitian ($K = \frac{B - B^*}{2}$).

Alternative notation:

• $A^*$ or $A^\dagger$ denotes the conjugate transpose (adjoint).
• In physics, Hermitian matrices are fundamental for quantum operators.