MS&E 201
Handout #0, Page of 1 of 6
Dynamic Systems
January 11, 2001
Problem Session Notes
Professor Edison Tse
A Review of Linear Algebra
Analysis of Dynamic Systems relies upon familiarity with a number of concepts from Linear Algebra and
Differential/Difference Equations.
This handout is a discussion of those Linear Algebra concepts which are
important in MS&E 201.
Linear Algebra
Matrix algebra is, among other things, a convenient way to represent systems of linear equations  the form in
which many dynamic systems problems are framed. Understanding matrix properties is therefore important to
the analysis of dynamic systems. To begin with, it is useful to be able to simplify the operation of raising a
square matrix to an arbitrary power:
A
k
where
A
is a square matrix and
k
is a nonnegative integer.
Definition.
A
diagonal matrix
is a square matrix for which
a
ij
= 0 for all
i
6
=
j
. Notice that it is a simple
thing to evaluate Λ
k
, where Λ is a diagonal matrix: the
i

th
diagonal element of Λ
k
is simply
λ
k
i
where
λ
i
is
the
i

th
diagonal element of Λ. Therefore it would be useful to be able to
diagonalize
a matrix, if possible,
for ease of computation of exponents of the matrix (and it provides insight as well, which is seen when one
considers the general solution to a dynamic system). In particular finding (invertible)
M
and Λ for a given
A
such that
A
=
M
Λ
M

1
(1)
would be desirable because
A
k
can then be expressed as
A
k
=
(
M
Λ
M

1
)(
M
Λ
M

1
)
. . .
(
M
Λ
M

1
)
=
M
Λ(
M

1
M
)Λ(
M

1
M
)
. . .
Λ
M

1
=
M
Λ
k
M

1
.
(2)
For (2) to apply we must have
AM
=
M
Λ. Rewriting
M
in terms of its column vectors
v
i
we obtain
A
v
1
v
2
. . .
v
n
=
v
1
v
2
. . .
v
n
Λ
The above can be rewritten as
Av
1
Av
2
. . .
Av
n
=
λ
1
v
1
λ
2
v
2
. . .
λ
n
v
n
So that the
M
and Λ that we are looking for are described by
Av
i
=
λ
i
v
i
for all
i
(
A

λ
i
I
)
v
i
= 0
for all
i.
(3)
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MS&E 201
Handout #0, Page of 2 of 6
Dynamic Systems
January 11, 2001
Problem Session Notes
Professor Edison Tse
If (
A

λ
i
I
) is an invertible matrix then
v
i
= 0. But this cannot be the case because
v
i
is a column vector of
M
, a nonsingular (invertible) matrix. Therefore (
A

λ
i
I
) cannot be invertible and we now have an expression
for determining
λ
i
:
det(
A

λ
i
I
) = 0
for all
i.
(4)
Definition.
The
eigenvalues of a square matrix
A
are the set of
λ
’s which satisfy (4). Associated with each
of these eigenvalues is an eigenvector which together satisfy (3). An eigenvalue and its associated eigenvector
together comprise an eigenpair.
M
is called the
modal matrix
of
A
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 EdisonTse
 Linear Algebra, Professor Edison Tse

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