Matrices: Operations and Eigenvalues

Matrices

Matrix Operations

• Addition/Subtraction: Matrices of the same dimension can be added or subtracted element-wise.
• Scalar Multiplication: Each element of the matrix is multiplied by a scalar.
• Matrix Multiplication:
  • Defined for matrices A (m x n) and B (n x p).
  • Resulting matrix C is of size (m x p).
  • Element C[i][j] is the dot product of the i-th row of A and the j-th column of B.
• Transposition: Flipping a matrix over its diagonal, changing rows to columns.
• Inverse: A matrix A has an inverse A⁻¹ if AA⁻¹ = A⁻¹A = I (identity matrix). Only square matrices can have inverses.

Eigenvalues and Eigenvectors

• Eigenvalue: A scalar λ such that for a square matrix A, there exists a non-zero vector v (eigenvector) satisfying Av = λv.
• Characteristic Polynomial: Det(A - λI) = 0, where I is the identity matrix. The roots give the eigenvalues.
• Eigenvectors: Found by substituting eigenvalues back into (A - λI)v = 0 and solving for v.
• Properties:
  • Eigenvalues can be real or complex.
  • The number of eigenvalues equals the size of the matrix.
  • Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Differential Equations

First-order Differential Equations

• Form: dy/dx = f(x, y).
• Types:
  • Separable: Can be expressed as g(y)dy = h(x)dx.
  • Linear: Can be written as dy/dx + P(x)y = Q(x).
• Solution Methods:
  • Separation of Variables: Separate and integrate both sides.
  • Integrating Factor: For linear equations, use μ(x) = e^(∫P(x)dx) to solve.

Systems of Differential Equations

• Form: A system of equations involving multiple interrelated functions.
• Matrix Form: dx/dt = Ax, where A is a matrix of coefficients and x is a vector of variables.
• Solution Methods:
  • Eigenvalue Method: Solve for eigenvalues and eigenvectors of matrix A.
  • Diagonalization: If A can be diagonalized, solutions can be expressed in terms of exponentials of eigenvalues.

Higher Order Differential Equations

• Form: n-th order equations can be expressed as a linear combination of derivatives of a function.
• Homogeneous: If the equation equals zero (e.g., y⁽ⁿ⁾ + pₙy⁽ⁿ⁻¹⁾ + ... + p₀y = 0).
• Non-homogeneous: If it equals a function (e.g., y⁽ⁿ⁾ + pₙy⁽ⁿ⁻¹⁾ + ... + p₀y = g(x)).
• Solution Methods:
  • Characteristic Equation: For homogeneous, find roots to form general solution.
  • Method of Undetermined Coefficients: For non-homogeneous, guess a particular solution.
  • Variation of Parameters: Another technique for finding particular solutions.

Matrices

Matrix Operations

• Matrices of identical dimensions can be added or subtracted element-wise.
• Scalar multiplication involves multiplying every element of the matrix by a scalar value.
• Matrix multiplication requires compatible matrices, resulting in a new matrix where each element is the dot product of corresponding row and column.
• The transposition of a matrix switches its rows with columns, effectively mirroring it over the main diagonal.
• An inverse of a square matrix A exists if multiplying A by its inverse (A⁻¹) results in the identity matrix (I).

Eigenvalues and Eigenvectors

• An eigenvalue (λ) corresponds to a non-zero vector (eigenvector) such that the equation Av = λv holds true for a square matrix A.
• The characteristic polynomial, Det(A - λI) = 0, helps to find eigenvalues where I is the identity matrix.
• Substituting eigenvalues back into (A - λI)v = 0 reveals the corresponding eigenvectors.
• Properties of eigenvalues include the possibility of real or complex values and an equal count to the matrix size.
• Eigenvectors related to different eigenvalues are linearly independent.

Differential Equations

First-order Differential Equations

• The general form is expressed as dy/dx = f(x, y).
• Types include:
  • Separable equations, restructured as g(y)dy = h(x)dx for integration.
  • Linear equations, in the form dy/dx + P(x)y = Q(x).
• Solution methods encompass:
  • Separation of variables to isolate and integrate each side separately.
  • Using an integrating factor, μ(x) = e^(∫P(x)dx), to rearrange linear equations for solving.

Systems of Differential Equations

• Systems consist of multiple interrelated equations representing functions.
• Represented in matrix form as dx/dt = Ax, where A includes the coefficients and x is the variable vector.
• Eigenvalue methods provide solutions for such systems by calculating matrix A's eigenvalues and eigenvectors.
• Diagonalization allows for expressible solutions through exponentials of eigenvalues, if A is diagonalizable.

Higher Order Differential Equations

• n-th order equations are combinations of derivatives of a function.
• Classified into:
  • Homogeneous equations, which equal zero, e.g., y⁽ⁿ⁾ + pₙy⁽ⁿ⁻¹⁾ +...+ p₀y = 0.
  • Non-homogeneous equations, expressed as an equal function, e.g., y⁽ⁿ⁾ + pₙy⁽ⁿ⁻¹⁾ +...+ p₀y = g(x).
• Solution methods include:
  • The characteristic equation to find roots for general solutions in homogeneous cases.
  • The method of undetermined coefficients for guessing particular solutions in non-homogeneous cases.
  • Variation of parameters as an additional technique for finding particular solutions.