-
@ Leonardo Araujo
2025-02-22 02:53:44
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. They are foundational element in many areas of [[Mathematics]] and [[Engineering]], including [[Electronics]], [[Computer Science]], [[Finances]], and more.
### Notation and Terms
- **Dimensions**: The size of a matrix is defined by its number of rows and columns and is often referred to as `m x n`, where `m` is the number of rows and `n` is the number of columns.
- **Square Matrix**: A matrix with the same number of rows and columns (`n x n`).
- **Diagonal Matrix**: A square matrix where all elements off the main diagonal are zero.
- **Identity Matrix**: A diagonal matrix where all the elements on the main diagonal are 1. It's denoted as `I`.
- **Zero Matrix**: A matrix all of whose entries are zero.
### Basic Matrix Operations
1. **Addition and Subtraction**
- Matrices must be of the same dimensions to be added or subtracted.
- Add or subtract corresponding elements.
- Example:
- $$
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
+
\begin{bmatrix}
5 & 6 \\
7 & 8
\end{bmatrix}
=
\begin{bmatrix}
6 & 8 \\
10 & 12
\end{bmatrix}
$$
2. **Scalar Multiplication**
- Multiply every element of a matrix by a scalar (a single number).
- Example:
- $$
2 \times
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
=
\begin{bmatrix}
2 & 4 \\
6 & 8
\end{bmatrix}
$$
3. **Matrix Multiplication**
- The number of columns in the first matrix must be equal to the number of rows in the second matrix.
- The product of an `m x n` matrix and an `n x p` matrix is an `m x p` matrix.
- Multiply rows by columns, summing the products of the corresponding elements.
- Example:
- $$
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
\times
\begin{bmatrix}
2 & 0 \\
1 & 2
\end{bmatrix}
=
\begin{bmatrix}
(1 \times 2 + 2 \times 1) & (1 \times 0 + 2 \times 2) \\
(3 \times 2 + 4 \times 1) & (3 \times 0 + 4 \times 2)
\end{bmatrix}
=
\begin{bmatrix}
4 & 4 \\
10 & 8
\end{bmatrix}
$$
### Special Matrix Operations
1. **Determinant**
- Only for square matrices.
- A scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix.
- Example for a 2x2 matrix:
- $$
\text{det}
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
= ad - bc
$$
2. **Inverse**
- Only for square matrices.
- The matrix that, when multiplied by the original matrix, results in the identity matrix.
- Not all matrices have inverses; a matrix must be "nonsingular" to have an inverse.
-
![[Attachments/71ec50617bc4cc58018cf33db28167ab_MD5.png]]
### Practical Applications
- **Solving Systems of Linear Equations**
- Matrices are used to represent and solve systems of linear equations using methods like Gaussian elimination.$$X=A^{-1}\times B$$
- **Transformations in Computer Graphics**
- Matrix multiplication is used to perform geometric transformations such as rotations, translations, and scaling.$$R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$$
##### Example System of Linear Equations
Suppose we have the following system of linear equations:
$$
\begin{align*}
3x + 4y &= 5 \\
2x - y &= 1
\end{align*}
$$
This system can be expressed as a matrix equation $AX=B$ where:
- $A$ is the matrix of coefficients,
- $X$ is the column matrix of variables,
- $B$ is the column matrix of constants.
* ***Matrix A** (coefficients): $\begin{bmatrix} 3 & 4 \\ 2 & -1 \end{bmatrix}$
* ***Matrix X** (variables): $\begin{bmatrix} x \\ y \end{bmatrix}$
* ***Matrix B** (constants): $\begin{bmatrix} 5 \\ 1 \end{bmatrix}$
Now Organising in Matrix form
$$\begin{bmatrix} 3 & 4 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}$$
##### Solving the Equation
To solve for $X$, we can calculate the inverse of $A$ (provided $A$ is invertible) and then multiply it by $B$:
$$X=A^{-1}\times B$$
## Matrices with SymPy
```python
from sympy import Matrix, symbols
# Define symbols
x, y, z = symbols('x y z')
# Define a 2x2 matrix
A = Matrix([[1, 2], [3, 4]])
print("Matrix A:")
print(A)
# Define a 3x3 matrix with symbolic elements
B = Matrix([[x, y, z], [y, z, x], [z, x, y]])
print("\nMatrix B:")
print(B)
# Define two matrices of the same size
C = Matrix([[5, 6], [7, 8]])
D = Matrix([[1, 1], [1, 1]])
# Addition
E = C + D
print("\nMatrix Addition (C + D):")
print(E)
# Subtraction
F = C - D
print("\nMatrix Subtraction (C - D):")
print(F)
# Scalar multiplication
G = 2 * A
print("\nScalar Multiplication (2 * A):")
print(G)
# Matrix multiplication
H = A * C
print("\nMatrix Multiplication (A * C):")
print(H)
# Determinant of a matrix
det_A = A.det()
print("\nDeterminant of Matrix A:")
print(det_A)
# Inverse of a matrix
inv_A = A.inv()
print("\nInverse of Matrix A:")
print(inv_A)
# Define the coefficient matrix A and the constant matrix B
A_sys = Matrix([[3, 4], [2, -1]])
B_sys = Matrix([5, 1])
# Solve the system AX = B
X = A_sys.inv() * B_sys
print("\nSolution to the system of linear equations:")
print(X)
# Compute eigenvalues and eigenvectors of a matrix
eigenvals = A.eigenvals()
eigenvects = A.eigenvects()
print("\nEigenvalues of Matrix A:")
print(eigenvals)
print("\nEigenvectors of Matrix A:")
print(eigenvects)
```
## References
* [Dear linear algebra students, This is what matrices (and matrix manipulation) really look like](https://www.youtube.com/watch?v=4csuTO7UTMo)
* [Essence of linear algebra](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)
* [The Applications of Matrices | What I wish my teachers told me way earlier](https://www.youtube.com/watch?v=rowWM-MijXU)
* [Inverse of 2x2 Matrix](https://www.cuemath.com/algebra/inverse-of-2x2-matrix/)
* [Matrices Tutorial](https://www.cuemath.com/algebra/solve-matrices/)