Find eigenvalues and eigenvectors of a $2 \\times 2$ matrix.
", "licence": "Creative Commons Attribution 4.0 International"}, "allQuestions": true, "shuffleQuestions": false, "questions": [], "percentPass": 0, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "feedback": {"showtotalmark": true, "advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true, "enterreviewmodeimmediately": false, "showexpectedanswerswhen": "never", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Eigenvalues and Eigenvectors of a 2x2 Matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Hayley Bishop", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/93/"}], "functions": {}, "ungrouped_variables": ["test1", "a21", "a22", "b1", "mna", "mnb", "s2", "s1", "a12", "test", "a11", "that", "da", "a1", "c2", "c1", "mxb", "mxa", "a", "b", "tra", "n", "this", "s"], "tags": ["diagonalising matrices", "eigenvalues", "eigenvalues of matrix", "eigenvectors of matrix", "matrix", "matrix eigenvalues", "test5"], "advice": "\\[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 $\\mathrm{trace}(A) = \\var{trA},\\;\\;\\;\\mathrm{det}(A)=\\var{dA}$
1. $\\lambda_1=\\var{mnA}$
\nWe 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.
\n\\[\\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}\\]
\nThis gives the two equations:
\n\\[ \\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 $y_1=\\var{-s*(a11-mnA)}$
\nHence the eigenvector we want is \\[\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mnA)} \\end{pmatrix}\\]
\n2. $\\lambda_2=\\var{mxA}$
\nIn this case we have the equations:
\n\\[ \\begin{eqnarray*} \\simplify[std]{{a11-mxA}x + {a12}y}&=&0\\\\ \\simplify[std]{{a21}x + {a22-mxA}y}&=&0 \\end{eqnarray*} \\]
\nOnce again there is only one equation, so putting $x=1$ in the first equation we get $y_2=\\var{-s*(a11-mxA)}$
\nHence the eigenvector we want is \\[\\begin{pmatrix} 1 \\\\ \\var{-s*(a11-mxA)} \\end{pmatrix}\\]
", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n \n \nFind the eigenvalues of $A$.
\n \n \n \nLet $a_1$ be the least eigenvalue of $A,\\;\\;\\; a_1=\\;\\;$[[0]]
\n \n \n \nLet $a_2$ be the greatest eigenvalue of $A,\\;\\; a_2=\\;\\;$[[1]]
\n \n \n \n \n \n \n ", "marks": 0, "gaps": [{"allowFractions": false, "scripts": {}, "maxValue": "{mnA}", "minValue": "{mnA}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "{mxA}", "minValue": "{mxA}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "Find eigenvectors for $A$.
\nLet $(1,y_1)^\\mathrm{T}$ be an eigenvector corresponding to the eigenvalue $a_1$.
\n$y_1=$ [[0]]
\nLet $(1,y_2)^\\mathrm{T}$ be an eigenvector corresponding to the eigenvalue $a_2$.
\n$y_2=$ [[1]]
", "marks": 0, "gaps": [{"allowFractions": false, "scripts": {}, "maxValue": "{s*(mnA-a11)}", "minValue": "{s*(mnA-a11)}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "scripts": {}, "maxValue": "{s*(mxA-a11)}", "minValue": "{s*(mxA-a11)}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "Find the eigenvalues and eigenvectors for the matrix $A$ where:
\\[ A=\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22} \\end{pmatrix}\\]
21/04/13 Based on Bill's \"Eigenvalues and eigenvectors\" combo question...
", "description": "Find eigenvalues and eigenvectors of $A$ $2 \\times 2$ matrix.
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