Matrix Reduction and Diagonalization

Posted on Nov 18, 2024 in Mathematics

ITEM 4: Determinants, Matrix Reduction, and Diagonalization

QR Factorization Theorem

If A is an m × n matrix with linearly independent columns, then A can be factored as A = QR, where Q is a matrix with orthonormal columns and R is an upper triangular matrix.

Example: Finding a QR Factorization

Find a QR factorization for the matrix A = [1 -2 1 / -1 3 2 / 1 1 -4].

Solution

By applying the Gram-Schmidt process to the columns of A, we get:

  • q1 = [1/√3, -1/√3, 1/√3]T
  • q2 = [0, 1/√2, 1/√2]T
  • q3 = [2/√6, 1/√6, -1/√6]T

Therefore, Q = [1/√3, 0, 2/√6 / -1/√3, 1/√2, 1/√6 / 1/√3, 1/√2, -1/√6] and R = QTA = [√3, -2√3-5/3√3 / 0 √2-2√3 / 0 0 1/3√6].

Calculating A120

Given P-1AP = D, we have PP-1APP-1 = PDP-1, which simplifies to A = PDP-1. Then A2 = PDP-1PDP-1 = PD2P-1 (since PP-1 = I). Therefore, A120 = PD120P-1.

ITEM 5: Matrix Reduction and Diagonalization

5.1. Generalized Eigenspace

Example

We will seek the generalized eigenspace of the matrix A = (1 -2 3 -4 / 0 1 1 2 / 0 0 1 4 / 0 0 0 -3).

The eigenvalues with algebraic multiplicities are:

  • λ1 = 1, m(λ1) = 3
  • λ2 = -3, m(λ2) = 1

Calculations for generalized eigenspaces follow (details omitted for brevity).

Then Eg(A, λ1) = Ker(I3 – A)3.

For the second eigenvalue, Eg(A, λ2) = E(A, λ2) = Ker(-3I3 – A).

5.2. Primary Decomposition Theorem (up to the same multiplicity)

Examples: Finding Generalized Eigenvector Basis

(Further examples and calculations omitted for brevity)

5.3. Vector Recurrences

(Content omitted for brevity)

5.4. Homogeneous Difference Equations

Examples

(Examples and solutions to difference equations omitted for brevity)