Find characteristic polynomial, eigenvalues and eigenvectors of a 2x2 matrix

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Find characteristic polynomial, eigenvalues and eigenvectors of a 2x2 matrix

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Julia Goedecke 2 years, 9 months ago

Julia Goedecke 2 years, 9 months ago

Julia Goedecke 2 years, 9 months ago

Julia Goedecke 2 years, 11 months ago

Julia Goedecke 2 years, 11 months ago

Julia Goedecke 2 years, 11 months ago

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Julia Goedecke 2 years, 9 months ago

Julia Goedecke 2 years, 9 months ago

Julia Goedecke 2 years, 9 months ago

Julia Goedecke 2 years, 11 months ago

Julia Goedecke 2 years, 11 months ago

Julia Goedecke 2 years, 11 months ago

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Created this as a copy of WBQ 9.3 Characteristic poly, eigenvalues and eigenvectors 2x2 - randomised.

Name	Status	Author	Last Modified
Invertibility via eigenvalues	Ready to use	Julia Goedecke	07/04/2022 14:34
Find characteristic polynomial, eigenvalues and eigenvectors of a 2x2 matrix	Ready to use	Julia Goedecke	21/09/2022 11:48

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Question statement

Find the characteristic polynomial, eigenvalues and eigenvectors (bases for eigenspaces) for the matrix $\var[fractionNumbers]{matrixA}$ .

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			Name	Type	Generated Value

			a		Testing condition not satisfied
			b		Testing condition not satisfied

			Name	Type	Generated Value

			Name	Type	Generated Value

			chiA		Testing condition not satisfied
			chiAexpanded		Testing condition not satisfied

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			Name	Type	Generated Value

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random(-5..5)

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Picking between 0 as eigenvalue and a non-zero eigenvalue

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RREFa
chiA
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matrixA

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Char poly Gap-fill
1. Gap char_A Mathematical expression
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eigenvalues Gap-fill
1. Gap eigenvalues List of numbers
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evectors Gap-fill
1. Gap Chosen eigenvalue Number entry
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2. Gap Eigenvector(s) Matrix entry
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Advice

The characteristic polynomial of $A$ is defined as $\chi_A(t)=\det(tI-A)$. So in this case you should calculate

\[\begin{vmatrix} \simplify{t-{matrixA[0][0]}} & \var{-matrixA[0][1]} \\ \var{-matrixA[1][0]} & \simplify{t-{matrixA[1][1]}} \end{vmatrix}=\simplify[!zeroterm]{(t-{matrixA[0][0]})(t-{matrixA[1][1]})-{matrixA[0][1]*matrixA[1][0]}}=\var{chiAexpanded}= \var[timesdot]{chiA}.\]

You can use the formula for the determinant a $2\times 2$ matrix:$ \begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$.

The eigenvalues are exactly the roots of the characteristic polynomial. So the eigenvalues of $A$ are $\var{a}$ and $\var{b}$.

To calculate the eigenvectors for a given eigenvalue $\lambda$, we calculate the kernel of $A-\lambda I$.

So, for $\lambda=\var{a}$, we calculate the kernel of $\var{matrixA-a*id(2)}$, using the Gauss-Jordan algorithm.

\[ \begin{pmatrix} \var{matrixA[0][0]-a} & \var{matrixA[0][1]} \\ \var{matrixA[1][0]}& \var{matrixA[1][1]-a} \end{pmatrix} \to \begin{pmatrix}1 & \var[fractionNumbers]{matrixA[0][1]/(matrixA[0][0]-a)} \\ \var{matrixA[1][0]}& \var{matrixA[1][1]-a} \end{pmatrix} \to \var[fractionNumbers]{RREFa}\] \[ \begin{pmatrix} \var{matrixA[0][0]-a} & \var{matrixA[0][1]} \\ \var{matrixA[1][0]}& \var{matrixA[1][1]-a} \end{pmatrix} \to \begin{pmatrix}0 & 1\\ \var{matrixA[1][0]}& \var{matrixA[1][1]-a} \end{pmatrix} \to \var[fractionNumbers]{RREFa} \]

The reduced row echelon form is $\var[fractionNumbers]{RREFa}$. If we write the eigenvector as $\begin{pmatrix}x\\y\end{pmatrix}$, this means that we have $\simplify[fractionNumbers, timesdot,all]{{RREFa[0][0]}*x+{RREFa[0][1]}*y=0}$. We can choose $y=1$, which gives us $x=\var[fractionNumbers]{-RREFa[0][1]}$. But $x$ can be any number, and we can choose it to be $1$. So the eigenvector for $\lambda=\var{a}$ is $\var[fractionNumbers]{v1}$ (or any multiple of this).

And for $\lambda=\var{b}$, the reduced row echelon form of $\var{matrixA-b*id(2)}$ is $\var[fractionNumbers]{RREFb}$. If we write the eigenvector as $\begin{pmatrix}x\\y\end{pmatrix}$ again, this means that we have $\simplify[fractionNumbers, timesdot,all]{{RREFb[0][0]}*x+{RREFb[0][1]}*y=0}$. We can choose $y=1$, which gives us $x=\var[fractionNumbers]{-RREFb[0][1]}$. But $x$ can be any number, and we can choose it to be $1$. So the eigenvector is $\var[fractionNumbers]{v2}$ (or any multiple).

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Testing condition not satisfied

Testing condition not satisfied

Testing condition not satisfied

testinverse

Testing condition not satisfied

Error evaluating variable detp: Variable P is not defined.

Testing condition not satisfied