Eigenvalues and Eigenvectors of a 2x2 Matrix

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Question statement

Find the eigenvalues and eigenvectors for the matrix $A$ where:

$A=\begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22} \end{pmatrix}$

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			Name	Type	Generated Value

test1	integer	-8
a21	number	2
a22	integer	-1
b1	integer	6
mna	integer	-3
mnb	integer	4
s2	integer	-1
s1	integer	1
a12	integer	-1
test	integer	-4
a11	integer	-4
that	integer	2
da	number	6
a1	integer	4
c2	integer	6
c1	integer	-3
mxb	integer	6
mxa	integer	-2
a	integer	-3
b	integer	-2
tra	integer	-5
n	integer	6
this	integer	3
s	integer	-1

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Find the eigenvalues of $A$ .

Let $a_1$ be the least eigenvalue of $A,\;\;\; a_1=\;\;$ Gap 0.

Let $a_2$ be the greatest eigenvalue of $A,\;\; a_2=\;\;$ Gap 1.

Advice

Matrix $A$

$A - \lambda I_2 = \begin{pmatrix} \var{a11}-\lambda & \var{a12}\\ \var{a21} & \var{a22}-\lambda \end{pmatrix}$
Hence the characteristic polynomial

$p(\lambda)$ is:

$\begin{eqnarray*} \mathrm{det}\left(A-\lambda I_2 \right)&=&\simplify[zeroTerm]{({a11}-lambda)({a22}-lambda)-{a12}*{a21}}\\ &=& \simplify[std]{lambda^2-{trA}*lambda+{dA}}\\ &=&\simplify[std]{(lambda-{a})(lambda-{b})} \end{eqnarray*}$
We see that on solving

$p(\lambda)=0$ we get the eigenvalues:

$\lambda_1=\var{mnA},\;\;\;\lambda_2=\var{mxA}$
Note: We could have found the characteristic polynomial by noting that for a 2 × 2 matrix

$A$ then the characteristic polynomial is

$\lambda^2-\mathrm{trace}(A)+\mathrm{det}(A)$
where

$\mathrm{trace}(A) = \var{trA},\;\;\;\mathrm{det}(A)=\var{dA}$

Finding the eigenvectors:

1. $\lambda_1=\var{mnA}$

We have the eigenspace is given by all $v=(x,y)^\mathrm{T}$ such that $(\simplify{A-{mnA}}I_2)v=(0,0)^\mathrm{T}$ i.e.

$\begin{pmatrix} \var{a11-mnA}&\var{a12}\\ \var{a21}&\var{a22-mnA} \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} =\begin{pmatrix} 0 \\ 0 \end{pmatrix}$

This gives the two equations:

$\begin{eqnarray*} \simplify[std]{{a11-mnA}x + {a12}y}&=&0\\ \simplify[std]{{a21}x + {a22-mnA}y}&=&0 \end{eqnarray*}$
There is only one equation here as we see that the equations are the same (one is a multiple of the other).

So putting $x=1$ in the first equation we get $y_1=\var{-s*(a11-mnA)}$

Hence the eigenvector we want is

$\begin{pmatrix} 1 \\ \var{-s*(a11-mnA)} \end{pmatrix}$

2. $\lambda_2=\var{mxA}$

In this case we have the equations:

$\begin{eqnarray*} \simplify[std]{{a11-mxA}x + {a12}y}&=&0\\ \simplify[std]{{a21}x + {a22-mxA}y}&=&0 \end{eqnarray*}$

Once again there is only one equation, so putting $x=1$ in the first equation we get $y_2=\var{-s*(a11-mxA)}$

Hence the eigenvector we want is

$\begin{pmatrix} 1 \\ \var{-s*(a11-mxA)} \end{pmatrix}$

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Eigenvalues and Eigenvectors of a 2x2 Matrix

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Taxonomy: mathcentre

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Christian Lawson-Perfect 10 years ago

Hayley Bishop 11 years, 7 months ago

Error in variable testing condition

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Generated value: `integer`

Parts

Gap-fill

Matrix $A$

Finding the eigenvectors:

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Eigenvalues and Eigenvectors of a 2x2 Matrix

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 10 years ago

Hayley Bishop 11 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Matrix AA

Finding the eigenvectors:

Generated value: `integer`

Matrix $A$