MA1034: Process Modelling using Linear Algebra

Session 06: Diagonalization

Prof. Alejandro Ucan-Puc

Tecnológico de Monterrey • Departamento de Matemáticas

Goals of the Session

  • Understand change of basis matrices $P_{B \to C}$.
  • Formulate the conditions for matrix diagonalization $A = P D P^{-1}$.
  • Apply matrix powers $A^k = P D^k P^{-1}$ to model multi-stage industrial dynamics.

Motivation: Automobile Assembly Line

An automobile assembly line has 3 workstations (Body, Paint, Assembly) with production rates interacting via:

$$A = \begin{pmatrix} 1.1 & 0.2 & 0.1 \\ 0.3 & 0.9 & 0.2 \\ 0.1 & 0.1 & 1.0 \end{pmatrix}$$

By switching to an eigenvector basis $B = \{v_1, v_2, v_3\}$ representing decoupled production trade-offs, $A$ becomes diagonal!

Change of Basis Matrix

Definition: Let $B = \{v_1, \dots, v_n\}$ and $C = \{w_1, \dots, w_n\}$ be two bases for $V$. The change of basis matrix from $B$ to $C$ is:

$$P_{C \leftarrow B} = \begin{pmatrix} [v_1]_C & [v_2]_C & \dots & [v_n]_C \end{pmatrix}$$

Satisfying: $[v]_C = P_{C \leftarrow B} [v]_B$.

Matrix Diagonalization Theorem

Definition: An $n \times n$ matrix $A$ is diagonalizable if it is similar to a diagonal matrix $D$, i.e., there exists an invertible matrix $P$ such that:

$$A = P D P^{-1} \quad \text{or} \quad D = P^{-1} A P$$

Theorem: $A$ is diagonalizable if and only if $A$ has $n$ linearly independent eigenvectors.

  • Columns of $P$ are the eigenvectors $v_1, \dots, v_n$.
  • Diagonal entries of $D$ are the eigenvalues $\lambda_1, \dots, \lambda_n$.

Powers of Diagonalizable Matrices

To compute the state after $k$ production cycles $A^k$:

$$A^k = (P D P^{-1})^k = P D^k P^{-1}$$

where $D^k = \begin{pmatrix} \lambda_1^k & & 0 \\ & \ddots & \\ 0 & & \lambda_n^k \end{pmatrix}$.

Summary & Next Step

Diagonalization simplifies matrix operations by expressing operators in an optimal eigen-coordinate system.

Next Session: Orthogonality, Inner Products, and Least Squares.