Eigenvalues & eigenvectors of a 2x2

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Name	Status	Author	Last Modified
Eigenvalues & eigenvectors of a 2x2	Ready to use	Frank Doheny	14/03/2018 21:04
Owen's copy of Eigenvalues & eigenvectors of a 2x2	draft	Owen Jepps	15/05/2017 09:17
Elias Jakobus's copy of Owen's copy of Eigenvalues & eigenvectors of a 2x2	draft	Elias Jakobus Willemse	23/05/2017 12:58
Eigenvalues & eigenvectors	draft	Matthew Mears	27/03/2020 16:32
Eigenvalues and one eigenvector of a 2x2 matrix	draft	Peter Johnston	16/04/2020 21:49
Timothy's copy of Eigenvalues and one eigenvector of a 2x2 matrix	draft	Timothy Gould	16/04/2019 06:05
Maria's copy of Eigenvalues and one eigenvector of a 2x2 matrix	draft	Maria Aneiros	23/05/2019 06:32
Eigenvalues/Eigenvector of a 2x2 matrix	draft	Thomas Waters	12/09/2019 20:04

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Question statement

Given the matrix

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

Give any introductory information the student needs.

			Name	Type	Generated Value

a11	integer	9
c1	integer	17
a12	integer	-24
k	integer	3
a21	integer	1
a22	integer	20
lambda1	integer	12
lambda2	integer	17

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1. Gap Gap 0. Number entry
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Part b) Gap-fill
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Calculate the eigenvalues of the matrix A

$\lambda_1$ is the lesser of the two eigenvalues and $\lambda_2$ is the greater of the two eigenvalues;

$\lambda_1$ = Gap 0.

$\lambda_2$ = Gap 1.

Advice

The eigenvalues of a matrix are the values of $\lambda$ that satisfy the relation

$|A-\lambda I| = 0$

$\begin{vmatrix} \var{a11}-\lambda&\var{a12}\\ \var{a21}&\var{a22}-\lambda\\ \end{vmatrix}=0$

This gives:

$(\var{a11}-\lambda)*(\var{a22}-\lambda)-(\var{a12})*(\var{a21})=0$

$\lambda^2-\simplify{{a11}+{a22}}\lambda+\simplify{{a11}*{a22}-{a21}*{a12}}=0$

This can be solved using factorisation or by formula to give:

$\lambda =\var{lambda1}$ and $\lambda =\var{lambda2}$

An eigenvector $v=\begin{pmatrix} x\\ y\\ \end{pmatrix}$ corresponding to an eigenvalue $\lambda$ must satisfy the relation: $(A-\lambda I)v = 0$

so for $\lambda=\var{lambda1}$

$\begin{pmatrix} \simplify{{a11}-{lambda1}}&\var{a12}\\ \var{a21}&\simplify{{a22}-{lambda1}}\\ \end{pmatrix}\begin{pmatrix} x\\ \var{a21}\\ \end{pmatrix}=0$

thus

$\var{a21}x+\simplify{{a22}-{lambda1}}*\var{a21}=0$

$\var{a21}x=-\simplify{({a22}-{lambda1})*{a21}}$

$x=-\simplify{({a22}-{lambda1})}$

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This question is used in the following exams:

Matrix arithmetic by Frank Doheny in Matrices.
My vector exam by Chris Graham in Chris's workspace.
more sophisticated mathematics examples by Josh Lim in NUMBAS workshop demo.

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Frank Doheny 7 years, 11 months ago

Frank Doheny 7 years, 11 months ago

Frank Doheny 8 years ago

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Eigenvalues & eigenvectors of a 2x2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Frank Doheny 7 years, 11 months ago

Frank Doheny 7 years, 11 months ago

Frank Doheny 8 years ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`