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@ Leonardo Araujo
2025-02-22 03:28:20A 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, wheremis the number of rows andnis 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
- Addition and Subtraction
- Matrices must be of the same dimensions to be added or subtracted.
- Add or subtract corresponding elements.
- Example:
$$\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}+\begin{bmatrix}5 & 6 \\7 & 8\end{bmatrix}=\begin{bmatrix}6 & 8 \\10 & 12\end{bmatrix}$$
- Scalar Multiplication
- Multiply every element of a matrix by a scalar (a single number).
- Example:
$$ 2 \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix} $$- 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 nmatrix and ann x pmatrix is anm x pmatrix. - 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
- 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:
$$ \text{det} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc $$
- 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.
Practical Applications
- Solving Systems of Linear Equations
- Matrices are used to represent and solve systems of linear equations using methods like Gaussian elimination.
$$X=A^{-1}\times B$$
- Matrices are used to represent and solve systems of linear equations using methods like Gaussian elimination.
- Transformations in Computer Graphics
- Matrix multiplication is used to perform geometric transformations such as rotations, translations, and scaling.
$$R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$$
- Matrix multiplication is used to perform geometric transformations such as rotations, translations, and scaling.
Example System of Linear Equations
Suppose we have the following system of linear equations:
$$3x + 4y = 5\\2x - y = 1$$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
- Dimensions: The size of a matrix is defined by its number of rows and columns and is often referred to as