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

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|
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|Since &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a triangular matrix, the eigenvalues are the entries on the diagonal.
 +
|-
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|Hence, the only eigenvalue of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is &nbsp;<math style="vertical-align: 0px">5.</math>
 
|}
 
|}
  
 
{| 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, to find a basis for the eigenspace corresponding to &nbsp;<math style="vertical-align: -4px">5,</math>&nbsp; we need to solve &nbsp;<math style="vertical-align: -5px">(A-5I)\vec{x}=\vec{0}.</math>
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|-
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|To do this, we use row reduction. Thus, we get
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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\displaystyle{A-5I} & = & \displaystyle{\begin{bmatrix}
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          5 & 1 \\
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          0 & 5
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        \end{bmatrix}-\begin{bmatrix}
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          5 & 0 \\
 +
          0 & 5
 +
        \end{bmatrix}}\\
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&&\\
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& \sim & \displaystyle{\begin{bmatrix}
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          0 & 1 \\
 +
          0 & 0
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        \end{bmatrix}.}
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\end{array}</math>
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|-
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|Solving this system, we see &nbsp;<math style="vertical-align: -4px">x_1</math>&nbsp; is a free variable and &nbsp;<math style="vertical-align: -4px">x_2=0.</math>
 +
|-
 +
|Therefore, a basis for this eigenspace is
 
|-
 
|-
 
|
 
|
 +
::<math>\bigg\{\begin{bmatrix}
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          1  \\
 +
          0 
 +
        \end{bmatrix}\bigg\}.</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
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|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; The only eigenvalue of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is &nbsp;<math style="vertical-align: 0px">5</math>&nbsp; and the corresponding eigenspace has basis &nbsp;<math style="vertical-align: -20px">\bigg\{\begin{bmatrix}
 +
          1  \\
 +
          0 
 +
        \end{bmatrix}\bigg\}.</math>
 +
 
 
|-
 
|-
 
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
 
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
 
|}
 
|}
 
[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
 
[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 14:56, 13 October 2017

Let  

(a) Find a basis for the eigenspace(s) 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.}

(b) Is 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}   diagonalizable? Explain.


Foundations:  
Recall:
1. The eigenvalues of a triangular matrix are the entries on the diagonal.
2. By the Diagonalization Theorem, an  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 n\times n}   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}   is diagonalizable
if and only 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 A}   has  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 n}   linearly independent eigenvectors.


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 A}   is a triangular matrix, the eigenvalues are the entries on the diagonal.
Hence, the only eigenvalue 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}   is  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.}
Step 2:  
Now, to find a basis for the eigenspace corresponding to  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,}   we need to solve  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-5I)\vec{x}=\vec{0}.}
To do this, we use row reduction. Thus, we get
       
Solving this system, we see  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 x_1}   is a free variable 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 x_2=0.}
Therefore, a basis for this eigenspace is
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 \bigg\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}\bigg\}.}

(b)

Step 1:  
Step 2:  


Final Answer:  
   (a)     The only eigenvalue 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}   is  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}   and the corresponding eigenspace has basis  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 \bigg\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}\bigg\}.}
   (b)    

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