For, $s$ , $t$ real numbers, and $A$ a 3x3 matrix, find the set of 3-vectors such that $sA^{2} textbf{v} = tA...











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Statement of the question: "Let $A$ be the real matrix:



begin{bmatrix} 2 3 1 \ 1 8 2 \ 3 0 2 end{bmatrix}



For real numbers $s$, $t$, define $V(s,t) = { textbf{v} in mathbb{R}^{3} : sA^{2}textbf{v} = tAtextbf{v} }$



For all $(s,t)$ such that $V neq {0}$, find the dimension of $V$. Also, find a basis for $V$. $0$ denotes the zero-vector."



In previous parts, I have found bases for each of the eigenspaces for the eigenvalues $2, sqrt{15}, -sqrt{15}$ and have shown that the map $m : textbf{v} rightarrow Atextbf{v}$ is a bijection.



My attempt to solve this problem was to simply let $sA^{2}textbf{v} = tAtextbf{v}$ for some $s, t in mathbb{R}, textbf{v} in mathbb{R}^{3}$ and to then solve the resulting simultaneous equations, but that is either impossible or I have made some mistake at that step.



Any hints as to how this problem can be solved are appreciated. Thanks.










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  • Find the eigenvalues of $s A^2-t A$. If $V neq 0$ then one of these eigenvalues must be $0$
    – Lozenges
    Nov 17 at 17:16















up vote
0
down vote

favorite












Statement of the question: "Let $A$ be the real matrix:



begin{bmatrix} 2 3 1 \ 1 8 2 \ 3 0 2 end{bmatrix}



For real numbers $s$, $t$, define $V(s,t) = { textbf{v} in mathbb{R}^{3} : sA^{2}textbf{v} = tAtextbf{v} }$



For all $(s,t)$ such that $V neq {0}$, find the dimension of $V$. Also, find a basis for $V$. $0$ denotes the zero-vector."



In previous parts, I have found bases for each of the eigenspaces for the eigenvalues $2, sqrt{15}, -sqrt{15}$ and have shown that the map $m : textbf{v} rightarrow Atextbf{v}$ is a bijection.



My attempt to solve this problem was to simply let $sA^{2}textbf{v} = tAtextbf{v}$ for some $s, t in mathbb{R}, textbf{v} in mathbb{R}^{3}$ and to then solve the resulting simultaneous equations, but that is either impossible or I have made some mistake at that step.



Any hints as to how this problem can be solved are appreciated. Thanks.










share|cite|improve this question






















  • Find the eigenvalues of $s A^2-t A$. If $V neq 0$ then one of these eigenvalues must be $0$
    – Lozenges
    Nov 17 at 17:16













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Statement of the question: "Let $A$ be the real matrix:



begin{bmatrix} 2 3 1 \ 1 8 2 \ 3 0 2 end{bmatrix}



For real numbers $s$, $t$, define $V(s,t) = { textbf{v} in mathbb{R}^{3} : sA^{2}textbf{v} = tAtextbf{v} }$



For all $(s,t)$ such that $V neq {0}$, find the dimension of $V$. Also, find a basis for $V$. $0$ denotes the zero-vector."



In previous parts, I have found bases for each of the eigenspaces for the eigenvalues $2, sqrt{15}, -sqrt{15}$ and have shown that the map $m : textbf{v} rightarrow Atextbf{v}$ is a bijection.



My attempt to solve this problem was to simply let $sA^{2}textbf{v} = tAtextbf{v}$ for some $s, t in mathbb{R}, textbf{v} in mathbb{R}^{3}$ and to then solve the resulting simultaneous equations, but that is either impossible or I have made some mistake at that step.



Any hints as to how this problem can be solved are appreciated. Thanks.










share|cite|improve this question













Statement of the question: "Let $A$ be the real matrix:



begin{bmatrix} 2 3 1 \ 1 8 2 \ 3 0 2 end{bmatrix}



For real numbers $s$, $t$, define $V(s,t) = { textbf{v} in mathbb{R}^{3} : sA^{2}textbf{v} = tAtextbf{v} }$



For all $(s,t)$ such that $V neq {0}$, find the dimension of $V$. Also, find a basis for $V$. $0$ denotes the zero-vector."



In previous parts, I have found bases for each of the eigenspaces for the eigenvalues $2, sqrt{15}, -sqrt{15}$ and have shown that the map $m : textbf{v} rightarrow Atextbf{v}$ is a bijection.



My attempt to solve this problem was to simply let $sA^{2}textbf{v} = tAtextbf{v}$ for some $s, t in mathbb{R}, textbf{v} in mathbb{R}^{3}$ and to then solve the resulting simultaneous equations, but that is either impossible or I have made some mistake at that step.



Any hints as to how this problem can be solved are appreciated. Thanks.







real-analysis linear-algebra matrices






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asked Nov 17 at 16:08









David Hughes

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  • Find the eigenvalues of $s A^2-t A$. If $V neq 0$ then one of these eigenvalues must be $0$
    – Lozenges
    Nov 17 at 17:16


















  • Find the eigenvalues of $s A^2-t A$. If $V neq 0$ then one of these eigenvalues must be $0$
    – Lozenges
    Nov 17 at 17:16
















Find the eigenvalues of $s A^2-t A$. If $V neq 0$ then one of these eigenvalues must be $0$
– Lozenges
Nov 17 at 17:16




Find the eigenvalues of $s A^2-t A$. If $V neq 0$ then one of these eigenvalues must be $0$
– Lozenges
Nov 17 at 17:16















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