Determinants, Eigenvalues, and Linear Dynamical Systems in Linear Algebra
Determinants and Invertibility
What is the Determinant of a Matrix?
The determinant of a matrix is a scalar value that provides information about the matrix’s invertibility. Only square matrices have determinants.
Calculating Determinants
For a 2×2 matrix A = [a b; c d], the determinant is calculated as det(A) = ad – bc.
For 3×3 matrices and larger, methods like cofactor expansion or row reduction are used.
Determinants and Row Operations
Elementary row operations affect the determinant as follows:
- Interchanging two rows: The determinant changes sign.
- Multiplying a row by a constant U: The determinant is multiplied by U.
- Adding a multiple of one row to another: The determinant remains unchanged.
Product Theorem for Determinants
The determinant of the product of two matrices is the product of their determinants: det(AB) = det(A) * det(B).
This implies that det(A^k) = (det(A))^k for any positive integer k.
Invertibility and Determinants
A square matrix A is invertible if and only if its determinant is non-zero (det(A) ≠ 0).
For an invertible matrix A, the determinant of its inverse is the reciprocal of its determinant: det(A^-1) = 1/det(A).
Orthogonal Matrices
A matrix A is orthogonal if its inverse is equal to its transpose: A^-1 = A^T.
The determinant of an orthogonal matrix is either 1 or -1.
Eigenvalues and Eigenvectors
Definition
An eigenvalue of a square matrix A is a scalar λ such that there exists a non-zero vector x (eigenvector) satisfying the equation Ax = λx.
Finding Eigenvalues and Eigenvectors
To find eigenvalues, we solve the characteristic equation det(λI – A) = 0, where I is the identity matrix.
For each eigenvalue, we solve the system of equations (λI – A)x = 0 to find the corresponding eigenvectors.
Diagonalizable Matrices
Definition
A square matrix A is diagonalizable if there exists an invertible matrix P such that P^-1AP is a diagonal matrix.
Diagonalization Criteria
A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, where n is the size of the matrix.
Diagonalization Process
To diagonalize a matrix A:
- Find the eigenvalues and eigenvectors of A.
- Form the matrix P using the eigenvectors as columns.
- Calculate the inverse of P (P^-1).
- The diagonal matrix D is obtained as D = P^-1AP.
Linear Dynamical Systems
Definition
A linear dynamical system is a sequence of vectors v0, v1, v2, … governed by the recurrence relation vk+1 = Avk, where A is the migration matrix and v0 is the initial vector.
Dominant Eigenvalue
The dominant eigenvalue of a matrix is the eigenvalue with the largest absolute value, provided it has multiplicity 1 (appears only once).
Exact Formula for Vectors
For a diagonalizable matrix A with eigenvalues λ1, λ2, …, λn and corresponding eigenvectors x1, x2, …, xn, the k-th vector in the dynamical system can be expressed as:
vk = c1 * λ1^k * x1 + c2 * λ2^k * x2 + … + cn * λn^k * xn
where c1, c2, …, cn are constants determined by the initial vector v0.