Homework 4 - Due: 02/15
Problem 1
In class last week, we discussed changes to the LTV system
For this problem
- Extend the coordinate change to the full system:
- Show that if
is a fundamental matrix for the system where is invertible then - Using this result re-derive the variation of constants formula
where .
Problem 2
Prove that the Euclidean norm
Hint: Use the Cauchy-Schwarz inequality
Problem 3
Let
Show that if
is an eigenvalue of and is a corresponding eigenvector, then and is an eigenvector (you can assume for simplicity). In other words, eigenvalues of symmetric matrices are always real and eigenvectors can always be chosen to be real.Hint: Show that
is real.Show that eigenvectors of
corresponding to distinct eigenvalues are orthogonal.
Problem 4
Let
We define the adjoint of
Prove that if
Hint: Let
Problem 5
Using the stability definitions given in class, determine if the systems below are stable, asymptotically stable, globally asymptotically stable, or neither. The first two systems are in
and and if and otherwise
Justify your answers using only the definitions of stability (not eigenvalues or Lyapunov’s method).
Problem 6
First, some definitions.
Given a linear operator
, a subspace is called -invariant if .For a linear system
on , this means that implies (reason: ).If
is an eigenvector of with a real eigenvalue, then is a 1-dimensional invariant subspace. For a Jordan block, the eigenvector and the generalized eigenvectors together span an invariant subspace.The case of a pair of complex eigenvalues was discussed in Problem Set 2, Problem 3
The sum of all invariant subspaces corresponding to eigenvalues with
is called the stable invariant subspace; the corresponding object for is the unstable invariant subspace; together these two subspaces span .
Consider the LTI system