Difference between revisions of "031 Review Part 3, Problem 6"

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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1:    
 
!Step 1:    
 +
|-
 +
|Since &nbsp;<math>\vec{x}</math>&nbsp; is an eigenvector of &nbsp;<math>A</math>&nbsp; corresponding to the eigenvalue &nbsp;<math>2,</math>&nbsp; we know &nbsp;<math>\vec{x}\neq \vec{0}</math>&nbsp; and
 
|-
 
|-
 
|
 
|
 +
::<math>A\vec{x}=2\vec{x}.</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
+
|Now, we have
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math>\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}}\\
 +
&&\\
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& = & \displaystyle{5\vec{x}}.
 +
\end{array}</math>
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|-
 +
|Hence, since &nbsp;<math>\vec{x}\ne \vec{0},</math>&nbsp; we conclude that &nbsp;<math>\vec{x}</math>&nbsp; is an eigenvector of &nbsp;<math>A^3-A^2+I</math>&nbsp; corresponding to the eigenvalue &nbsp;<math>5.</math>
 +
 
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|Since &nbsp;<math>\vec{y}</math>&nbsp; is an eigenvector of &nbsp;<math>A</math>&nbsp; corresponding to the eigenvalue &nbsp;<math>3,</math>&nbsp; we know &nbsp;<math>\vec{y}\neq \vec{0}</math>&nbsp; and
 
|-
 
|-
 
|
 
|
 +
::<math>A\vec{y}=3\vec{y}.</math>
 +
|-
 +
|Also, since &nbsp;<math>A</math>&nbsp; is invertible, &nbsp;<math>A^{-1}</math>&nbsp; exists.
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 +
|-
 +
|Now, we multiply the equation from Step 1 on the left by &nbsp;<math>A^{-1}</math> to obtain
 +
|-
 +
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{A^{-1}(A\vec{y})} & = & \displaystyle{A^{-1}(3\vec{y}}\\
 +
&&\\
 +
& = & \displaystyle{3(A^{-1}\vec{y}).}
 +
\end{array}</math>
 +
|-
 +
|Now, we have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\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}}\\
 +
&&\\
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& = & \displaystyle{\vec{y}.}
 +
\end{array}</math>
 +
|-
 +
|Hence, &nbsp;<math>A^{-1}\vec{y}=\frac{1}{3}\vec{y}.</math>
 +
|-
 +
|Therefore, &nbsp;<math>\vec{y}</math>&nbsp; is an eigenvector of &nbsp;<math>A^{-1}</math> corresponding to the eigenvalue &nbsp;<math>\frac{1}{3}.</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; See solution above.
 
|-
 
|-
−
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; See solution above.
 
|}
 
|}
 
[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
 
[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 08:27, 11 October 2017

(a) Show that if    is an eigenvector of the matrix    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
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\vec{x}=\lambda\vec{x}.}


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
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\vec{x}=2\vec{x}.}
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
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\vec{y}=3\vec{y}.}
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.

Return to Sample Exam

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