System Dynamics (24.509)
II. Mathematical Background
Review of Matrix Algebra and Matrix Calculus
Elementary Operations
Given:
Addition
Scalar multiplication
Matrix multiplication
where, in general, 
.
Matrix times a vector
Inner product
Transpose
Also note that
Determinants
Given 
, the determinant of 
is denoted as 
or 
2 x 2 matrix
n x n matrix
(use Laplace's Expansion)
where the cofactor, 
, of the ij element of the original matrix is given as
and the minor, 
, of the 
element in the 
matrix is the determinant of the matrix formed by deleting the ith row and the jth column from the original matrix. As an example, consider a general 
matrix,
Expanding along row one, we have
and
Note also that if 
is in triangular form (i.e. all the elements below or above the diagonal are zero), then the determinant of 
is simply the product of the diagonal terms.
Matrix Inverse
The inverse matrix is defined by the relation 
, where 
is the identity matrix and 
is called the inverse of 
. A simple formula can be written in terms of the adjoint matrix and determinant, or
where the adjoint matrix is formed by replacing each element of a matrix by its cofactor and then taking the transpose. Note that if 
and 
are square matrices, then
Proof:
Matrix Equations
Given the matrix equation, 
, the solution vector 
can be written as
Two important classes of problems arise, depending on the value of 
:
1. If 
, the system is
exists. This implies that the det
and that the rows and columns of 
are linearly independent.
2. If 
, the system is
is singular. A singular matrix is one whose determinant is zero. This means that the rows and columns of 
are linearly dependent.
Eigenvalues and Eigenvectors
If we let 
in a standard system of algebraic equations, then one has
The value 
is an
matrix 
if there is a non-zero vector 
such that eqn. (2.27) is valid. In this case, 
is an eigenvector of the matrix 
corresponding to eigenvalue 
. Equation (2.27) is often rewritten as
In this form, it becomes apparent that for a non-trivial solution, we must require that
This expression is called the characteristic equation. The roots of the characteristic polynomial gives n values of 
(not necessarily distinct). Once the n eigenvalues are known, the ith eigenvector can be determined from eqn. (2.28) by substituting 
. There will be one eigenvector corresponding to each eigenvalue.
The eigenvalues and eigenvectors of a matrix are extremely important for describing the dynamic behavior of time dependent systems. If one knows all the eigenvalues and eigenvectors of a linear, stationary, lumped parameter system, then the system's complete time domain and frequency domain behavior can be simulated analytically, the stability characteristics of the system can be determined, the sensitivity of the system to changes in input parameters can be approximated, and so on. In fact, one can describe almost every aspect of this class of dynamic systems with the eigenvalues and eigenvectors of the system matrix.
One should note that
similar matrices have identical eigenvalues. Two matrices,
and 
, are said to be similar if they satisfy the similarity transformation
A particularly useful similarity transformation involves the diagonalization of a matrix. For any square matrix with distinct eigenvalues, we have
or
where 
is defined as the
and the matrix 
is simply a diagonal matrix containing the n distinct eigenvalues of 
as the diagonal terms. These matrices are usually written as
and
where 
is the ith eigenvector which corresponds to eigenvalue 
. A quick proof of eqn. (2.31.a) can be given as follows:
and premultiplying by 
gives eqn. (2.31a).
Matrix Calculus
Given:
Differentiation
Integration
Product Rules
Differentiation of Inverse Matrix
Proof: Starting with 
, one has
and solving this expression for 
gives the desired result.
Differentiation of a Determinant (time independent)
Recalling Laplace's Expansion
where 
is the cofactor of element ij, one has
Matrix Exponential
Some dynamic systems have analytic solutions that can be written in terms of the so-called
matrix exponential. The matrix exponential,
, is defined in terms of an infinite Taylor series expansion,
This is an analogous definition to its scalar counterpart,
The derivative and integral of 
, where 
is a
Derivative
Proof: Take the derivative of each term in eqn. (2.39), giving
Integral
Proof: Let 
, then
Functions of Square Matrix
In addition to the infinite series expansion for 
given in eqn. (2.39), one can find a closed form expansion for the matrix exponential. In fact, a general closed-form result can be obtained for any polynomial function of a square matrix.
Sylvester's Theorem (Distinct Roots):
If 
is any polynomial of the square matrix , and if 
represents one of the n (distinct) eigenvalues of , then
or equivalently,
An important application of Sylvester's theorem is in finding a closed-form solution for the matrix exponential.
Example 2.6 illustrates this process for the case of a
order system with distinct roots.
If the eigenvalues of the system matrix are not all distinct, then an alternate form of eqn. (2.42) is required.
Sylvester's Theorem (Repeated Roots):
If 
is any polynomial of the square matrix 
and if an eigenvalue is repeated s times, then
where
with
and if all the eigenvalues are equal, then 
.
In the limit of no repeated roots, the confluent form of Sylvester's theorem reduces to the standard form.
Example 2.7 illustrates the use of eqns. (2.44) - (2.46) for the case of a
order system with repeated roots.
|
|
 |
 |
24.509 Lecture Notes by Dr. J. R. White, UMass-Lowell (Spring 1997).
Return to Online Courses