031 Review Part 3, Problem 6

(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  $A$  corresponding to the eigenvalue 2, then  ${\vec {x}}$  is an eigenvector of  $A^{3}-A^{2}+I.$  What is the corresponding eigenvalue?

(b) Show that if  ${\vec {y}}$  is an eigenvector of the matrix  $A$  corresponding to the eigenvalue 3 and  $A$  is invertible, then  ${\vec {y}}$  is an eigenvector of  $A^{-1}.$  What is the corresponding eigenvalue?

Foundations:
An eigenvector  ${\vec {x}}$  of a matrix  $A$  corresponding to the eigenvalue  $\lambda$  is a nonzero vector such that
$A{\vec {x}}=\lambda {\vec {x}}.$

Solution:

(a)

Step 1:
Since  ${\vec {x}}$  is an eigenvector of  $A$  corresponding to the eigenvalue  $2,$  we know  ${\vec {x}}\neq {\vec {0}}$  and
$A{\vec {x}}=2{\vec {x}}.$

Step 2:
Now, we have
${\begin{array}{rcl}\displaystyle {(A^{3}-A^{2}+I){\vec {x}}}&=&\displaystyle {A^{3}{\vec {x}}-A^{2}{\vec {x}}+I{\vec {x}}}\\&&\\&=&\displaystyle {A\cdot A\cdot A{\vec {x}}-A\cdot A{\vec {x}}+{\vec {x}}}\\&&\\&=&\displaystyle {A\cdot A\cdot 2{\vec {x}}-A\cdot 2{\vec {x}}+{\vec {x}}}\\&&\\&=&\displaystyle {2A\cdot A{\vec {x}}-2A{\vec {x}}+{\vec {x}}}\\&&\\&=&\displaystyle {2A\cdot 2{\vec {x}}-2\cdot 2{\vec {x}}+{\vec {x}}}\\&&\\&=&\displaystyle {(2\cdot 2)A{\vec {x}}-4{\vec {x}}+{\vec {x}}}\\&&\\&=&\displaystyle {(4)\cdot 2{\vec {x}}-4{\vec {x}}+{\vec {x}}}\\&&\\&=&\displaystyle {5{\vec {x}}}.\end{array}}$
Hence, since  ${\vec {x}}\neq {\vec {0}},$  we conclude that  ${\vec {x}}$  is an eigenvector of  $A^{3}-A^{2}+I$  corresponding to the eigenvalue  $5.$

(b)

Step 1:
Since  ${\vec {y}}$  is an eigenvector of  $A$  corresponding to the eigenvalue  $3,$  we know  ${\vec {y}}\neq {\vec {0}}$  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.

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