Difference between revisions of "031 Review Part 3, Problem 7"

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<span class="exam">Use the Diagonalization Theorem to find the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and a basis for each eigenspace.
 
<span class="exam">Use the Diagonalization Theorem to find the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and a basis for each eigenspace.
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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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
 
|-
 
|-
 
|
 
|
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::<math>\begin{bmatrix}
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          3 & 0 & 0 \\
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          0 & 4 &0\\
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          0 & 0 & 3
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        \end{bmatrix}</math>
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|-
 +
|is a diagonal matrix, the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; and &nbsp;<math style="vertical-align: -1px">4</math>&nbsp; by the Diagonalization Theorem.
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 +
|-
 +
|By the Diagonalization Theorem, a basis for the eigenspace corresponding
 +
|-
 +
|to the eigenvalue &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; is
 +
|-
 +
|
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::<math>\Bigg\{\begin{bmatrix}
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          3 \\
 +
          0 \\
 +
          1
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        \end{bmatrix},\begin{bmatrix}
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          -1 \\
 +
          -3\\
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          0
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        \end{bmatrix}\Bigg\}</math>
 +
|-
 +
|and a basis for the eigenspace corresponding to the eigenvalue &nbsp;<math style="vertical-align: -1px">4</math>&nbsp; is
 
|-
 
|-
 
|
 
|
 +
::<math>\Bigg\{\begin{bmatrix}
 +
          0 \\
 +
          1 \\
 +
          1
 +
        \end{bmatrix}\Bigg\}.</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|&nbsp;&nbsp; &nbsp; &nbsp;  
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|&nbsp; &nbsp; &nbsp; &nbsp; The eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; and &nbsp;<math style="vertical-align: -1px">4.</math>&nbsp;
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp; A basis for the eigenspace corresponding
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp; to the eigenvalue &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; is
 +
|-
 +
|
 +
::::<math>\Bigg\{\begin{bmatrix}
 +
          3 \\
 +
          0 \\
 +
          1
 +
        \end{bmatrix},\begin{bmatrix}
 +
          -1 \\
 +
          -3\\
 +
          0
 +
        \end{bmatrix}\Bigg\}</math>
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; and a basis for the eigenspace corresponding to the eigenvalue &nbsp;<math style="vertical-align: -1px">4</math>&nbsp; is
 +
|-
 +
|
 +
::::<math>\Bigg\{\begin{bmatrix}
 +
          0 \\
 +
          1 \\
 +
          1
 +
        \end{bmatrix}\Bigg\}.</math>
 
|}
 
|}
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[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
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[[031_Review_Part_3|'''<u>Return to Review Problems</u>''']]

Latest revision as of 13:03, 15 October 2017

Let  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=\begin{bmatrix} 3 & 0 & -1 \\ 0 & 1 &-3\\ 1 & 0 & 0 \end{bmatrix}\begin{bmatrix} 3 & 0 & 0 \\ 0 & 4 &0\\ 0 & 0 & 3 \end{bmatrix}\begin{bmatrix} 0 & 0 & 1 \\ -3 & 1 &9\\ -1 & 0 & 3 \end{bmatrix}.}

Use the Diagonalization Theorem to find the eigenvalues 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}   and a basis for each eigenspace.

Foundations:  
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.
In fact,  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=PDP^{-1},}   with  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 D}   a diagonal matrix, if and only if the columns 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 P}   are  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 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.}   In this case, the diagonal entries 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 D}   are eigenvalues 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}   that
correspond, respectively , to the eigenvectors in  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 P.}


Solution:

Step 1:  
Since
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}3&0&0\\0&4&0\\0&0&3\end{bmatrix}}}
is a diagonal matrix, the eigenvalues 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}   are  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}   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 4}   by the Diagonalization Theorem.
Step 2:  
By the Diagonalization Theorem, a basis for the eigenspace 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}   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} 3 \\ 0 \\ 1 \end{bmatrix},\begin{bmatrix} -1 \\ -3\\ 0 \end{bmatrix}\Bigg\}}
and a basis for the eigenspace 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 4}   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} 0 \\ 1 \\ 1 \end{bmatrix}\Bigg\}.}


Final Answer:  
        The eigenvalues 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}   are  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}   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 4.}  
        A basis for the eigenspace 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}   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} 3 \\ 0 \\ 1 \end{bmatrix},\begin{bmatrix} -1 \\ -3\\ 0 \end{bmatrix}\Bigg\}}
        and a basis for the eigenspace 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 4}   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} 0 \\ 1 \\ 1 \end{bmatrix}\Bigg\}.}

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