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.
real-analysis linear-algebra matrices
asked Nov 17 at 16:08
David Hughes
1086
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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.
real-analysis linear-algebra matrices
asked Nov 17 at 16:08
David Hughes
1086
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
add a comment |
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.
real-analysis linear-algebra matrices
asked Nov 17 at 16:08
David Hughes
1086
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
real-analysis linear-algebra matrices
asked Nov 17 at 16:08
David Hughes
1086
asked Nov 17 at 16:08
David Hughes
1086
asked Nov 17 at 16:08
David Hughes
1086
asked Nov 17 at 16:08
David Hughes
1086
asked Nov 17 at 16:08
David Hughes
1086
1086
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
add a comment |
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
add a comment |
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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