MATH 348 - Advanced Engineering Mathematics June 25, 2009
Homework 4, Summer 2009 Due: June 30 , 2009
Eigenvalues - Eigenvectors - Diagionalization - Spectral Decomposition - Applications
1. Find the eigenvalues and eigenvectors of the following matrix.
A=
401
−210
−201
2. Given,
A="3 1
−2 1 #.
Determine the eigenvalues and eigenfunctions associated with the system of differential equations dx
dt =A·x(t).
3. Given,
A=
4000
0400
0020
1002
.
If Ais diagonalizable, then determine Dand Passociated with its decomposition PDP−1. Do not find P−1.
4. Square matrices having columns whose entries sum to 1 are often called stochastic matrices. Those with only non-negative
entries, for some power, are called regular stochastic matrices. Given a random process, with an initial state x0, the
application of Pon x0discretely steps the process forward in time. That is xn+1 =Pxn=Pnx0, n = 1,2,3, . . . . If a
matrix is a regular stochastic matrix then there exists a steady-state vector qsuch that Pq=q. This vector determines the
long term probabilities associated with an arbitrary inital state x0. The sequence of states, {x0,x1,x2,...,xn+1}, is called
aMarkov Chain. Given the regular stochastic matrix:
P=".1.6
.9.4#.
(a) Show that the steady-state vector of Pis q=2
5
3
5t
.
(b) Find the matrices Dand Qsuch that P=QDQ−1. That is, diagonalize the matrix P.
(c) Show that lim
n→∞
Pnx0=qwhere x0= [x1, x2]tis an arbitrary vector in R2such that x1+x2= 1.
5. Recall the Pauli Spin Matrix from homework 1,
σ2=σy="0−i
i0#.
(a) Show that σyis self-adjoint. 1
(b) Find the orthogonal diagonalization of σy.2
(c) Show that σy=λ1x1xh
1+λ2x2xh
2, where x1and x2are the normalized eigenvectors from part (b).
1Recall that self-adjoint means that A=Ah=¯
At. See the linear algebra handout on ticc.mines.edu for the definitions.
2This will require you to normalize vectors that contain imaginary numbers. Generally to normalize a vector xwe makea new vector ˆ
x=x
|x|=x
√xtx.
However, when using vectors with imaginary entries we must use the adjoint in our definition of inner-product. That is, we take a normalized vector
to be, ˆ
x=x
|x|=x
√xhx. If you do not do this then then your inner-products will become zero and your normalized vectors will be undefined, which I
guess they should be in real-space.
1