Difference between revisions of "031 Review Part 3, Problem 10"
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<span class="exam">Show that if <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of the matrix product <math style="vertical-align: 0px">AB</math> and <math style="vertical-align: -5px">B\vec{x}\ne \vec{0},</math> then <math style="vertical-align: 0px">B\vec{x}</math> is an eigenvector of <math style="vertical-align: 0px">BA.</math> | <span class="exam">Show that if <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of the matrix product <math style="vertical-align: 0px">AB</math> and <math style="vertical-align: -5px">B\vec{x}\ne \vec{0},</math> then <math style="vertical-align: 0px">B\vec{x}</math> is an eigenvector of <math style="vertical-align: 0px">BA.</math> | ||
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| − | [[031_Review_Part_3|'''<u>Return to | + | [[031_Review_Part_3|'''<u>Return to Review Problems</u>''']] |
Latest revision as of 14:09, 15 October 2017
Show that if is an eigenvector of the matrix product 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 AB} 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 B\vec{x}\ne \vec{0},} 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 B\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 BA.}
| Foundations: |
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| 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} is a nonzero vector such that |
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| for some scalar 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.} |
Solution:
| Step 1: |
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| 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 AB,} 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 |
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| for some scalar 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.} |
| Using associativity of matrix multiplication, we have |
|
| 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{BA(B\vec{x})} & = & \displaystyle{B(A(B\vec{x}))}\\ &&\\ & = & \displaystyle{B(\lambda \vec{x})}\\ &&\\ & = & \displaystyle{\lambda B\vec{x}.} \end{array}} |
| 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 B\vec{x}\neq \vec{0},} we can 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 B\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 AB.} |
| Final Answer: |
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| See solution above. |