Complex Numbers: Definition and Properties

Definition and Notation

• A complex number is a number of the form a + bi, where:
  • a is the real part
  • b is the imaginary part
  • i is the imaginary unit, which satisfies i^2 = -1

Properties

• Complex numbers can be added and subtracted component-wise:
  • (a + bi) + (c + di) = (a + c) + (b + d)i
  • (a + bi) - (c + di) = (a - c) + (b - d)i
• Complex numbers can be multiplied using the distributive property:
  • (a + bi) * (c + di) = (ac - bd) + (ad + bc)i
• Complex numbers can be divided using the conjugate:
  • (a + bi) / (c + di) = ((a + bi) * (c - di)) / (c^2 + d^2)

Conjugate and Modulus

• The conjugate of a + bi is a - bi
• The modulus (or absolute value) of a + bi is √(a^2 + b^2)

Polar Form

• Complex numbers can be represented in polar form:
  • a + bi = r(cos(θ) + i*sin(θ)), where:
    • r is the modulus (radius)
    • θ is the argument (angle)

De Moivre's Theorem

• r(cos(θ) + i*sin(θ))^n = r^n(cos(nθ) + i*sin(nθ))
• Used to raise complex numbers to integer powers

Euler's Formula

• e^(iθ) = cos(θ) + i*sin(θ)
• Relates exponential and trigonometric functions

Complex Numbers

• A complex number is a number of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, which satisfies i^2 = -1.

Properties of Complex Numbers

• Complex numbers can be added and subtracted component-wise.
• When adding or subtracting complex numbers, the real parts are added/subtracted, and the imaginary parts are added/subtracted.
• Complex numbers can be multiplied using the distributive property.
• When multiplying complex numbers, the real and imaginary parts are multiplied separately.
• Complex numbers can be divided using the conjugate.
• When dividing complex numbers, the conjugate of the denominator is used to simplify the expression.

Conjugate and Modulus

• The conjugate of a + bi is a - bi.
• The modulus (or absolute value) of a + bi is √(a^2 + b^2).
• The modulus represents the distance from the origin to the complex number on the complex plane.

Polar Form

• Complex numbers can be represented in polar form: a + bi = r(cos(θ) + i*sin(θ)).
• r is the modulus (radius) and θ is the argument (angle).
• Polar form is used to visualize complex numbers on the complex plane.

De Moivre's Theorem

• De Moivre's Theorem states: r(cos(θ) + i*sin(θ))^n = r^n(cos(nθ) + i*sin(nθ)).
• The theorem is used to raise complex numbers to integer powers.
• It provides a formula for calculating the result of raising a complex number to a power.

Euler's Formula

• Euler's Formula states: e^(iθ) = cos(θ) + i*sin(θ).
• The formula relates exponential and trigonometric functions.
• It provides a connection between the exponential function and the sine and cosine functions.

Matrix Determinants

• The determinant of a matrix is a scalar value that describes the properties of the matrix.
• Notated as |A| or det(A).
• Has the following properties:
  • Multiplicativity: det(AB) = det(A) × det(B).
  • Scalar multiplication: det(cA) = c^n × det(A) for an n × n matrix A.

Linear Transformations

• A linear transformation is a function between vector spaces that preserves vector operations.
• Can be represented by a matrix, where the matrix product of the matrix and a vector represents the transformation of the vector.
• Has the following properties:
  • Linearity: T(av + bw) = aT(v) + bT(w).
  • Composition: The composition of two linear transformations is also a linear transformation.
• Types of linear transformations:
  • Isomorphism: A bijective linear transformation.
  • Endomorphism: A linear transformation from a vector space to itself.

Matrix Equations

• A matrix equation is an equation involving matrices, often used to solve systems of linear equations.
• Types of matrix equations:
  • Linear systems: AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
  • Homogeneous systems: AX = 0, where A is the coefficient matrix and X is the variable matrix.
• Solution methods:
  • Gaussian elimination.
  • LU decomposition.
  • Inverse matrix method.

Eigenvalues

• An eigenvalue is a scalar that represents how a linear transformation changes a vector.
• Eigenvalue equation: AX = λX, where A is the matrix, λ is the eigenvalue, and X is the eigenvector.
• Properties:
  • Scalar multiplication: If λ is an eigenvalue, then kλ is also an eigenvalue for any scalar k.
  • Linear independence: Eigenvectors corresponding to distinct eigenvalues are linearly independent.
• Applications:
  • Markov chains.
  • Image compression.
  • Data analysis.

Matrix Operations

• Matrix addition: A + B = [aij + bij].
• Matrix multiplication: AB = [Σ(aik × bkj)].
• Scalar multiplication: cA = [cai].
• Matrix transpose: AT = [aji].
• Matrix inverse: A^(-1) (if A is invertible).