Module 7 : Discrete State Space Models

Lecture 1 : Characteristic Equation, eigenvalues and eigen vectors

1. Characteristic Equation, eigenvalues and eigen vectors

For a discrete state space model, the characteristic equation is defined as

$\displaystyle \vert zI -A \vert = 0 $

The roots of the characteristic equation are the eigenvalues of matrix A.

1. If , i.e., A is nonsingular and $ \lambda_1$, $ \lambda_2$, $ \cdots$, $ \lambda_n$ are the eigenvalues of A, then, $ \frac{1}{\lambda_1}$, $ \frac{1}{\lambda_2}$, $ \cdots$, $ \frac{1}{\lambda_n}$ will be the eigenvalues of A-1.

2. Eigenvalues of A and AT are same when A is a real matrix.

3. If A is a real symmetric matrix then all its eigenvalues are real.

The $ n\times 1$ vector $ v_i$ which satisfies the matrix equation

$\displaystyle A v_i = \lambda_i v_i$ (1)

where $ \lambda_i, \;\; i=1,2, \cdots, n$ denotes the ith eigenvalue, is called the eigen vector of A associated with the eigenvalue $ \lambda_i$. If eigenvalues are distinct, they can be solved directly from equation (1).
Properties of eigen vectors

1. An eigen vector cannot be a null vector.

2. If $ v_i$ is an eigen vector of A then $ m v_i$ is also an eigen vector of A where m is a scalar.

3. If A has n distinct eigenvalues, then the n eigen vectors are linearly independent.

Eigen vectors of multiple order eigenvalues

When the matrix $ A$ an eigenvalue $ \lambda$ of multiplicity $ m$, a full set of linearly independent may not exist. The number of linearly independent eigen vectors is equal to the degeneracy $ d$ of $ \lambda I -A$. The degeneracy is defined as

$\displaystyle d = n - r $

where $ n$ is the dimension of $ A$ and $ r$ is the rank of $ \lambda I -A$. Furthermore,

$\displaystyle 1 \le d \le m $

2 Similarity Transformation and Diagonalization

Square matrices A and $ \bar{A}$ are similar if

The non-singular matrix P is called similarity transformation matrix. It should be noted that eigenvalues of a square matrix A are not altered by similarity transformation.
Diagonalization:
If the system matrix A of a state variable model is diagonal then the state dynamics are decoupled from each other and solving the state equations become much more simpler.