MATH 280, Spring 2002 MATH 280, Spring 2002

VI: Characteristic Equations, Diagonalization and Eigenvectors


The due date for this theme is Friday, April 26, at noon.

Reading assignment: Section 5.2, 5.3, 5.4

Background reading: Section 5.1


Using eigenvalues and eigenvectors, it is possible to find a very easy way to represent a matrix transformation from $\QTR{Bbb}{R}^{n}$ to itself. In order to make this work, for a linear transformation given by an $n\times n$ matrix $A$, one must be able to first find the eigenvalues and corresponding eigenvectors of $A$. In this theme, we will study ways to find eigenvalues and eigenvectors and use them to rewrite our matrix transformations.

What does it mean to say two matrices are similar? Find a matrix similar to
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and show that the two are similar. If $A$ and $B$ are arbitrary similar matrices, show that $\det A=\det B$.

Let $A$ be an $n\times n$ matrix. What is the characteristic equation of $A$? Suppose that $A$ has $n$ distinct eigenvalues. Explain why $A$ is diagonalizable?

If $A$ has fewer than $n$ distinct eigenvalues, then $A$ may or may not be diagonalizable. The following two examples will demonstrate this.

Consider the matrix
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Show that $A$ is diagonalizable by writing $A=PDP^{-1}$, with $P$ invertible and $D$ diagonal. Find $A^{5}$. Next, let
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Show that $B$ is not diagonalizable.

Consider the matrix $A$ in the previous paragraph. Find a basis $\QTR{cal}{B}$ so that the $\QTR{cal}{B}$-matrix of the transformation determined by $A$ is diagonal. Justify your answer.

For the last problem, we will return to the theory of Markov chains. I suggest that you work on this problem with only the members of your group, since your grades are really dependent on each of you. As we will see, you cannot really trust the rest of your classmates.

We saw in the last theme that the stochastic matrix of a Markov chain always has an eigenvalue of $1$. This fact is going to help us understand the ability of your classmates to repeat things accurately. In the following problem, the ``rumor'' is a very simple one that is either true or false, like ``today is my birthday.''

Someone in the class has just started a rumor about you. You don't have to worry, because the rumor is true (and it is not bad). Or, do you have to worry? Certainly, some people lie. Even if everyone means to tell the truth, they don't always hear accurately what was said. So, even in the best situations, people don't always get the rumor right.

Let's suppose that people accurately repeat what they are told with probability $p$, where $0<p<1$. Since they either repeat it accurately or inaccurately, what is the probability that they repeat it inaccurately? Suppose you heard the correct version of the rumor. What is the vector $v_{c}$ whose first component is the probability that you repeat the original rumor correctly and second component is the probability that you repeat the original rumor incorrectly? Now, suppose you heard an incorrect version of the rumor. What is the vector $v_{i}$ whose first component is the probability that you repeat the original rumor correctly and second component is the probability that you repeat the original rumor incorrectly? Find the stochastic matrix for spreading this rumor. [Be careful, you must really understand what the previous sentences mean to get the right matrix.] Find the steady-state vector for a Markov chain associated to this matrix. What does this vector mean? How comfortable are you that in the long run, the rumor that people hear will be true? Why?

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