Hi everyone my name is Claire Tomlin and
I'm a professor of Electrical
Engineering and computer sciences at
Berkeley and this is the 29th module in
a series that we're recording for the
course
EEE CS 221 a which is linear system
theory I'm gonna have a brief module
just to present something called the
spectral mapping theorem in this one and
it follows on from our previous module
where we presented the Jordan form of a
matrix so I'm going to start by just
recapping a Jordan form so if we had
matrix and I'm going to write down a
possible Jordan form J for that matrix
this is an example where we have two
repeated eigenvalues one at lambda 1 and
one at lambda 2 and the matrix is not
diagonalizable but it can be put into
Jordan form and this is just an example
again so it's a it's an example where we
have one Jordan block of size 3
associated to the eigenvalue at lambda 1
so there's the Jordan block of size 3
and those are ones and everything below
the diagonal is 0 1 Jordan block of size
2 so lambda 1 lambda 1 1 okay
again that's a zero and then I have
three repeated eigen values that lambda
2 and there's a Jordan block of size 2
and a Jordan block of size 1 lambda 2 ok
so so there's four Jordan blocks related
to the system the minimal polynomial
which we defined as sy hat a of s so
associated to actually it's not a it's J
associated to this matrix J we can just
read off immediately from from the
jordan form so the largest jordan block
associated to the eigenvalue at lambda 1
is 3 and the largest Jordan block
associated to the eigen value at lambda
2.is is 2 okay so that gives us the
minimum polynomial of course the
characteristic polynomial we can of
course read off right away
and that's just going to be s minus
lambda 1 to the fifth because
of the five eigenvalues at lambda 1 and
s minus lambda 2 to the third because of
the three eigenvalues that lambda two
okay so we also see from this as we
talked about from our knowledge of the
similarity transform that brought us
into the jordan form from a general
matrix a with this structure eigen with
this eigen space structure is that
there's four eigenvectors associated to
this matrix so there's four eigenvectors
one for each jordan block so two
eigenvectors associated to lambda one
and two associated to lambda two it's an
eight dimensional system so that means
there's four generalized eigenvectors
okay so brief recap of the jordan form
the function of now suppose we have a
function a matrix a function of a matrix
so some function f let's suppose that
function f is analytic on the spectrum
of j then we can calculate that function
of the matrix and as we presented last
day the function f of j can be
represented in terms of an operation on
the jordan form and it's given as
follows so that is also in this case in
this example it's an it's an eight by
eight matrix so f of j is also an eight
by eight matrix and it's a matrix it's a
block diagonal matrix so associated to
we have similar blocks associated to the
corresponding eigenvalues so there's my
blocks for lambda one blocks for lambda
two here's the one block associated to
that eigenvalue at lambda two okay and
then as we presented last day the form
of that function is f of lambda one on
the diagonals f of lambda one and then
on the super diagonals the the immediate
super diagonal we have F prime of lambda
one so it's the derivative of F with
respect to its argument evaluated at
lambda one F prime of lambda one and in
the upper right-hand corner we have
f double-prime of lambda 1 over 2
factorial okay
then everything else here is zero so
similarly now we'll have again f of
lambda 1 on the diagonals f of lambda 1
and f primed of lambda 1 on the super
diagonal 0 here f of lambda 2 F of
lambda 2 F prime evaluated at lambda 2 0
and then just F of lambda 2 everything
else is 0 so these are zeros okay so
it's an upper triangular matrix in
general and so the eigenvalues of this
matrix F of J are just the diagonal
entries and so we see immediately in
this Jordan form and the function of the
Jordan form that the eigenvalues of f of
j are just f of lambda i where lambda i
are the eigenvalues of the matrix J and
so this is this is true in general for
Jordan forms and of course in the
special case a diagonal form and this
leads us to the statement of the
spectral mapping theorem so the spectral
mapping theorem is just the statement of
this of this fact spectral mapping
theorem ok so remember the spectrum is
the set of eigenvalues of a matrix so
and we we denote Sigma the spectrum so
Sigma of f of J the eigenvalues of this
function f of the matrix is equal to f
of sigma of j ok so it's the set F of
Sigma of J so it's the set F of lambda I
where I ranges from 1 up to however many
eigenvalues we have and actually we use
the terminology so
different Sigma previously for that so
basically you take all of the
eigenvalues of j and you evaluate f of
those eigenvalues and that gives you the
eigenvalues of f of j okay so so we've
done a recap of the jordan form of a
matrix and then we've done a recap of
the functional form of that matrix so f
of j so we have a general form basically
once we're given the jordan form now and
we're given the function that we want to
calculate of that matrix we can write
down the function of the matrix by
inspection and then from that general
form we get out the spectral mapping
theorem which states that the
eigenvalues of f of j are the set of
eigenvalues that function evaluated on
the eigenvalues of the matrix J okay so
in now we're we're in the future modules
we're now going to use this remember
that the point about this Jordan form is
in general it gives us an easier way of
computing functions of the matrix and
the the main function that we've been
interested in is the function e to the a
T the the matrix exponential where a
represents the system dynamic matrix the
square matrix which represents the
system dynamics so now we can think
about how we might transform a to Jordan
form and then compute e to the J T using
a formulation like this okay thank you
very much
