Difference between revisions of "031 Review Part 3, Problem 6"
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!Step 1: | !Step 1: | ||
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| − | |Since <math>\vec{x}</math> is an eigenvector of <math>A</math> corresponding to the eigenvalue <math>2,</math> we know <math>\vec{x}\neq \vec{0}</math> and | + | |Since <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of <math style="vertical-align: 0px">A</math> corresponding to the eigenvalue <math style="vertical-align: -4px">2,</math> we know <math style="vertical-align: -5px">\vec{x}\neq \vec{0}</math> and |
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\end{array}</math> | \end{array}</math> | ||
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| − | |Hence, since <math>\vec{x}\ne \vec{0},</math> we conclude that <math>\vec{x}</math> is an eigenvector of <math>A^3-A^2+I</math> corresponding to the eigenvalue <math>5.</math> | + | |Hence, since <math style="vertical-align: -5px">\vec{x}\ne \vec{0},</math> we conclude that <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of <math style="vertical-align: -2px">A^3-A^2+I</math> corresponding to the eigenvalue <math style="vertical-align: 0px">5.</math> |
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!Step 1: | !Step 1: | ||
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| − | |Since <math>\vec{y}</math> is an eigenvector of <math>A</math> corresponding to the eigenvalue <math>3,</math> we know <math>\vec{y}\neq \vec{0}</math> and | + | |Since <math style="vertical-align: -3px">\vec{y}</math> is an eigenvector of <math style="vertical-align: 0px">A</math> corresponding to the eigenvalue <math style="vertical-align: -4px">3,</math> we know <math style="vertical-align: -5px">\vec{y}\neq \vec{0}</math> and |
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::<math>A\vec{y}=3\vec{y}.</math> | ::<math>A\vec{y}=3\vec{y}.</math> | ||
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| − | |Also, since <math>A</math> is invertible, <math>A^{-1}</math> exists. | + | |Also, since <math style="vertical-align: 0px">A</math> is invertible, <math style="vertical-align: 0px">A^{-1}</math> exists. |
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!Step 2: | !Step 2: | ||
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| − | |Now, we multiply the equation from Step 1 on the left by <math>A^{-1}</math> to obtain | + | |Now, we multiply the equation from Step 1 on the left by <math style="vertical-align: 0px">A^{-1}</math> to obtain |
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\end{array}</math> | \end{array}</math> | ||
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| − | |Hence, <math>A^{-1}\vec{y}=\frac{1}{3}\vec{y}.</math> | + | |Hence, <math style="vertical-align: -13px">A^{-1}\vec{y}=\frac{1}{3}\vec{y}.</math> |
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| − | |Therefore, <math>\vec{y}</math> is an eigenvector of <math>A^{-1}</math> corresponding to the eigenvalue <math>\frac{1}{3}.</math> | + | |Therefore, <math style="vertical-align: -3px">\vec{y}</math> is an eigenvector of <math style="vertical-align: 0px">A^{-1}</math> corresponding to the eigenvalue <math style="vertical-align: -12px">\frac{1}{3}.</math> |
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Revision as of 07:34, 11 October 2017
(a) Show that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}} is an eigenvector of the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} corresponding to the eigenvalue 2, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}} is an eigenvector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^3-A^2+I.} What is the corresponding eigenvalue?
(b) Show that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}} is an eigenvector of the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} corresponding to the eigenvalue 3 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is invertible, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}} is an eigenvector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{-1}.} What is the corresponding eigenvalue?
| Foundations: |
|---|
| An eigenvector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}} of a matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} corresponding to the eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} is a nonzero vector such that |
|
Solution:
(a)
| Step 1: |
|---|
| Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}} is an eigenvector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} corresponding to the eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2,} we know Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}\neq \vec{0}} and |
|
| Step 2: |
|---|
| Now, we have |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{(A^3-A^2+I)\vec{x}} & = & \displaystyle{A^3\vec{x}-A^2\vec{x}+I\vec{x}}\\ &&\\ & = & \displaystyle{A\cdot A\cdot A\vec{x}-A\cdot A\vec{x}+\vec{x}}\\ &&\\ & = & \displaystyle{A\cdot A \cdot 2\vec{x}-A\cdot 2\vec{x}+\vec{x}}\\ &&\\ & = & \displaystyle{2A\cdot A\vec{x}-2A\vec{x}+\vec{x}}\\ &&\\ & = & \displaystyle{2A\cdot 2\vec{x}-2\cdot 2\vec{x}+\vec{x}}\\ &&\\ & = & \displaystyle{(2\cdot 2)A\vec{x}-4\vec{x}+\vec{x}}\\ &&\\ & = & \displaystyle{(4)\cdot 2\vec{x}-4\vec{x}+\vec{x}}\\ &&\\ & = & \displaystyle{5\vec{x}}. \end{array}} |
| Hence, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}\ne \vec{0},} we conclude that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}} is an eigenvector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^3-A^2+I} corresponding to the eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5.} |
(b)
| Step 1: |
|---|
| Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}} is an eigenvector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} corresponding to the eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3,} we know Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}\neq \vec{0}} and |
|
| Also, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is invertible, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{-1}} exists. |
| Step 2: |
|---|
| Now, we multiply the equation from Step 1 on the left by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{-1}} to obtain |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{A^{-1}(A\vec{y})} & = & \displaystyle{A^{-1}(3\vec{y}}\\ &&\\ & = & \displaystyle{3(A^{-1}\vec{y}).} \end{array}} |
| Now, we have |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{3(A^{-1}\vec{y})} & = & \displaystyle{A^{-1}(A\vec{y})}\\ &&\\ & = & \displaystyle{(A^{-1}A)\vec{y}}\\ &&\\ & = & \displaystyle{I\vec{y}}\\ &&\\ & = & \displaystyle{\vec{y}.} \end{array}} |
| Hence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{-1}\vec{y}=\frac{1}{3}\vec{y}.} |
| Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}} is an eigenvector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{-1}} corresponding to the eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3}.} |
| Final Answer: |
|---|
| (a) See solution above. |
| (b) See solution above. |