1.15 Eigenvalues and eigenvectorsIB Maths: Applications and Interpretation HL: Flashcards

What these 13 flashcards ask

What equation defines an eigenvalue \lambda and eigenvector \mathbf{v}?
How do you find the eigenvalues of M?
What is the characteristic equation of a 2\times2 matrix?
What is the trace of \begin{pmatrix}a&b\\ c&d\end{pmatrix}?
How do you find an eigenvector for a known \lambda?
Is \mathbf{0} an eigenvector?
What is the diagonalisation of M?
What must be true of the eigenvalues for diagonalisation here?
What is M^n in terms of P and D?
What is D^n for D=\begin{pmatrix}\lambda1&0\\ 0&\lambda2\end{pmatrix}?
What are the eigenvalues of M^n?
What does an eigenvalue of 1 mean in a population model?
In \mathbf{x}n=c1\lambda1^n\mathbf{v}1+c2\lambda2^n\mathbf{v}2, what controls the long-term behaviour?

Exam questions on 1.15 Eigenvalues and eigenvectors

$M=\begin{pmatrix}4&1\\ 2&3\end{pmatrix}$ .
Find an eigenvector of $M$ corresponding to the eigenvalue $2$ .2 marks
$M=\begin{pmatrix}1&2\\ 3&0\end{pmatrix}$ .
Write down the eigenvalues of $M^5$ .2 marks
The populations, in thousands, of two towns $A$ and $B$ in year $n$ are $a_n$ and $b_n$ . Each year 20% of the people in town $A$ move to town $B$ and 10% of the people in town $B$ move to town $A$ . Nobody else moves, so $\begin{pmatrix}a_{n+1}\\ b_{n+1}\end{pmatrix}=M\begin{pmatrix}a_n\\ b_n\end{pmatrix}$ with $M=\begin{pmatrix}0.8&0.1\\ 0.2&0.9\end{pmatrix}$ .
Show that the eigenvalues of $M$ are $1$ and $0.7$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).