Mixing
Determine dimension of set of matrices given that each matrix has same eigenvalue 0 and eigenvector
$begingroup$
Let V be the set of all $A_{3×3}$ matrices, where all of them have eigenvalue 0 and corresponding eigenvector $y=(1,2,3)^T$.
Assuming we have showed that V is a vector space, determine the dimension of V.
From the definition we know that:
$x$ and $lambda$ are eigenvector and eigenvalue iff $(A-lambda I)x=0$ .
So for each matrix $A_i$ in our set$$
(A_i-0)
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
=0
$$
$$
A_i
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
=0
$$
$$
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
in N(A_i)
$$
From this we find that $Nullity(A_i)geq1 Rightarrow $
$Rank(A_i) = n-Nullity(A_i)=3-1=2 Rightarrow $
$Rank(A_i)=Dim(C(A_i)) leq 2$
However, I am not sure how can I proceed from here?
1) How can determine the dimension of the matrix $A_i$ for sure?
2) How is that connected to the dimension of entire vector space formed by such matrices?
linear-algebra vector-spaces eigenvalues-eigenvectors
asked Jan 15 at 19:38
Jill WhiteJill White
133
$endgroup$
add a comment |
$begingroup$
Let V be the set of all $A_{3×3}$ matrices, where all of them have eigenvalue 0 and corresponding eigenvector $y=(1,2,3)^T$.
Assuming we have showed that V is a vector space, determine the dimension of V.
From the definition we know that:
$x$ and $lambda$ are eigenvector and eigenvalue iff $(A-lambda I)x=0$ .
So for each matrix $A_i$ in our set$$
(A_i-0)
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
=0
$$
$$
A_i
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
=0
$$
$$
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
in N(A_i)
$$
From this we find that $Nullity(A_i)geq1 Rightarrow $
$Rank(A_i) = n-Nullity(A_i)=3-1=2 Rightarrow $
$Rank(A_i)=Dim(C(A_i)) leq 2$
However, I am not sure how can I proceed from here?
1) How can determine the dimension of the matrix $A_i$ for sure?
2) How is that connected to the dimension of entire vector space formed by such matrices?
linear-algebra vector-spaces eigenvalues-eigenvectors
asked Jan 15 at 19:38
Jill WhiteJill White
133
$endgroup$
add a comment |
$begingroup$
Let V be the set of all $A_{3×3}$ matrices, where all of them have eigenvalue 0 and corresponding eigenvector $y=(1,2,3)^T$.
Assuming we have showed that V is a vector space, determine the dimension of V.
From the definition we know that:
$x$ and $lambda$ are eigenvector and eigenvalue iff $(A-lambda I)x=0$ .
So for each matrix $A_i$ in our set$$
(A_i-0)
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
=0
$$
$$
A_i
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
=0
$$
$$
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
in N(A_i)
$$
From this we find that $Nullity(A_i)geq1 Rightarrow $
$Rank(A_i) = n-Nullity(A_i)=3-1=2 Rightarrow $
$Rank(A_i)=Dim(C(A_i)) leq 2$
However, I am not sure how can I proceed from here?
1) How can determine the dimension of the matrix $A_i$ for sure?
2) How is that connected to the dimension of entire vector space formed by such matrices?
linear-algebra vector-spaces eigenvalues-eigenvectors
asked Jan 15 at 19:38
Jill WhiteJill White
133
$endgroup$
Let V be the set of all $A_{3×3}$ matrices, where all of them have eigenvalue 0 and corresponding eigenvector $y=(1,2,3)^T$.
Assuming we have showed that V is a vector space, determine the dimension of V.
From the definition we know that:
$x$ and $lambda$ are eigenvector and eigenvalue iff $(A-lambda I)x=0$ .
So for each matrix $A_i$ in our set$$
(A_i-0)
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
=0
$$
$$
A_i
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
=0
$$
$$
begin{pmatrix}
1 \
2 \
3 \
end{pmatrix}
in N(A_i)
$$
From this we find that $Nullity(A_i)geq1 Rightarrow $
$Rank(A_i) = n-Nullity(A_i)=3-1=2 Rightarrow $
$Rank(A_i)=Dim(C(A_i)) leq 2$
However, I am not sure how can I proceed from here?
1) How can determine the dimension of the matrix $A_i$ for sure?
2) How is that connected to the dimension of entire vector space formed by such matrices?
linear-algebra vector-spaces eigenvalues-eigenvectors
linear-algebra vector-spaces eigenvalues-eigenvectors
asked Jan 15 at 19:38
Jill WhiteJill White
133
asked Jan 15 at 19:38
Jill WhiteJill White
133
asked Jan 15 at 19:38
Jill WhiteJill White
133
asked Jan 15 at 19:38
Jill WhiteJill White
133
asked Jan 15 at 19:38
Jill WhiteJill White
133
133
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3074858%2fdetermine-dimension-of-set-of-matrices-given-that-each-matrix-has-same-eigenvalu%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3074858%2fdetermine-dimension-of-set-of-matrices-given-that-each-matrix-has-same-eigenvalu%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
