Diagonalizability over $mathbb{C}$ and $mathbb{R}$ respectively
$begingroup$
I am new to Linear Algebra, and would love some feedback regarding the following question, which I found a bit confusing:
$$A = begin{Bmatrix}0&1&0&0\0&0&1&0\0&0&0&1\1&0&0&0end{Bmatrix}$$
Which of the following two statements is true and/or false?
$A$ is diagonalizable over $mathbb{C}$
$A$ is diagonalizable over $mathbb{R}$
I calculated the characteristic polynomial to be $λ^4+1=0$, and I doublechecked the calculations. This can of course be rewritten as $(λ-1)(λ+1)(λ^2+1)$.
This leads me to the conclusion that the eigenvalues over $mathbb{C}$ are $1, -1, i$, and $-i$, while the eigenvalues over $mathbb{R}$ are $1$ and $-1$.
I then calculated the eigenvectors, which came out to be:
for $λ=1: (1,1,1,1)$
for $λ=-1: (-1,1,-1,1)$
for $λ=i: (i,-1,-i,1)$
for $λ=-i: (-i,-1,i,1)$
Now, clearly the geometric multiplicity is 1 in each of these cases (while only the first two cases are applicable "over $mathbb{R}$"). I assume the algebraic multiplicity is also one in each case.
Therefore, $A$ is diagonalizable both over $mathbb{C}$ and over $mathbb{R}$.
Thank you!
complex-numbers real-numbers diagonalization
$endgroup$
add a comment |
$begingroup$
I am new to Linear Algebra, and would love some feedback regarding the following question, which I found a bit confusing:
$$A = begin{Bmatrix}0&1&0&0\0&0&1&0\0&0&0&1\1&0&0&0end{Bmatrix}$$
Which of the following two statements is true and/or false?
$A$ is diagonalizable over $mathbb{C}$
$A$ is diagonalizable over $mathbb{R}$
I calculated the characteristic polynomial to be $λ^4+1=0$, and I doublechecked the calculations. This can of course be rewritten as $(λ-1)(λ+1)(λ^2+1)$.
This leads me to the conclusion that the eigenvalues over $mathbb{C}$ are $1, -1, i$, and $-i$, while the eigenvalues over $mathbb{R}$ are $1$ and $-1$.
I then calculated the eigenvectors, which came out to be:
for $λ=1: (1,1,1,1)$
for $λ=-1: (-1,1,-1,1)$
for $λ=i: (i,-1,-i,1)$
for $λ=-i: (-i,-1,i,1)$
Now, clearly the geometric multiplicity is 1 in each of these cases (while only the first two cases are applicable "over $mathbb{R}$"). I assume the algebraic multiplicity is also one in each case.
Therefore, $A$ is diagonalizable both over $mathbb{C}$ and over $mathbb{R}$.
Thank you!
complex-numbers real-numbers diagonalization
$endgroup$
2
$begingroup$
$lambda^4 - 1 = (lambda^2 - 1)(lambda^2 + 1) = (lambda - 1)(lambda + 1)(lambda^2 + 1)$; $lambda^4 + 1$ is irreducible over $Bbb R$. Check again! Cheers!
$endgroup$
– Robert Lewis
Jan 18 at 8:26
add a comment |
$begingroup$
I am new to Linear Algebra, and would love some feedback regarding the following question, which I found a bit confusing:
$$A = begin{Bmatrix}0&1&0&0\0&0&1&0\0&0&0&1\1&0&0&0end{Bmatrix}$$
Which of the following two statements is true and/or false?
$A$ is diagonalizable over $mathbb{C}$
$A$ is diagonalizable over $mathbb{R}$
I calculated the characteristic polynomial to be $λ^4+1=0$, and I doublechecked the calculations. This can of course be rewritten as $(λ-1)(λ+1)(λ^2+1)$.
This leads me to the conclusion that the eigenvalues over $mathbb{C}$ are $1, -1, i$, and $-i$, while the eigenvalues over $mathbb{R}$ are $1$ and $-1$.
I then calculated the eigenvectors, which came out to be:
for $λ=1: (1,1,1,1)$
for $λ=-1: (-1,1,-1,1)$
for $λ=i: (i,-1,-i,1)$
for $λ=-i: (-i,-1,i,1)$
Now, clearly the geometric multiplicity is 1 in each of these cases (while only the first two cases are applicable "over $mathbb{R}$"). I assume the algebraic multiplicity is also one in each case.
Therefore, $A$ is diagonalizable both over $mathbb{C}$ and over $mathbb{R}$.
Thank you!
complex-numbers real-numbers diagonalization
$endgroup$
I am new to Linear Algebra, and would love some feedback regarding the following question, which I found a bit confusing:
$$A = begin{Bmatrix}0&1&0&0\0&0&1&0\0&0&0&1\1&0&0&0end{Bmatrix}$$
Which of the following two statements is true and/or false?
$A$ is diagonalizable over $mathbb{C}$
$A$ is diagonalizable over $mathbb{R}$
I calculated the characteristic polynomial to be $λ^4+1=0$, and I doublechecked the calculations. This can of course be rewritten as $(λ-1)(λ+1)(λ^2+1)$.
This leads me to the conclusion that the eigenvalues over $mathbb{C}$ are $1, -1, i$, and $-i$, while the eigenvalues over $mathbb{R}$ are $1$ and $-1$.
I then calculated the eigenvectors, which came out to be:
for $λ=1: (1,1,1,1)$
for $λ=-1: (-1,1,-1,1)$
for $λ=i: (i,-1,-i,1)$
for $λ=-i: (-i,-1,i,1)$
Now, clearly the geometric multiplicity is 1 in each of these cases (while only the first two cases are applicable "over $mathbb{R}$"). I assume the algebraic multiplicity is also one in each case.
Therefore, $A$ is diagonalizable both over $mathbb{C}$ and over $mathbb{R}$.
Thank you!
complex-numbers real-numbers diagonalization
complex-numbers real-numbers diagonalization
edited Jan 18 at 8:40
idriskameni
749321
749321
asked Jan 18 at 8:04
daltadalta
1578
1578
2
$begingroup$
$lambda^4 - 1 = (lambda^2 - 1)(lambda^2 + 1) = (lambda - 1)(lambda + 1)(lambda^2 + 1)$; $lambda^4 + 1$ is irreducible over $Bbb R$. Check again! Cheers!
$endgroup$
– Robert Lewis
Jan 18 at 8:26
add a comment |
2
$begingroup$
$lambda^4 - 1 = (lambda^2 - 1)(lambda^2 + 1) = (lambda - 1)(lambda + 1)(lambda^2 + 1)$; $lambda^4 + 1$ is irreducible over $Bbb R$. Check again! Cheers!
$endgroup$
– Robert Lewis
Jan 18 at 8:26
2
2
$begingroup$
$lambda^4 - 1 = (lambda^2 - 1)(lambda^2 + 1) = (lambda - 1)(lambda + 1)(lambda^2 + 1)$; $lambda^4 + 1$ is irreducible over $Bbb R$. Check again! Cheers!
$endgroup$
– Robert Lewis
Jan 18 at 8:26
$begingroup$
$lambda^4 - 1 = (lambda^2 - 1)(lambda^2 + 1) = (lambda - 1)(lambda + 1)(lambda^2 + 1)$; $lambda^4 + 1$ is irreducible over $Bbb R$. Check again! Cheers!
$endgroup$
– Robert Lewis
Jan 18 at 8:26
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
All your work seems correct to me, but $A$ is not diagonalisable over $mathbb{R}$, since your eigenvectors aren't over $mathbb{R}$, but over $mathbb{C}$. The diagonalisation would be $PDP^{-1}=A$ where:
$$
P=begin{pmatrix}
1 & -1 & i & -i \
1 & 1 & -1 & -1 \
1 & -1 & -i & i \
1 & 1 & 1 & 1
end{pmatrix},
D=operatorname{diag}(1, -1, i, -i)
$$
And since $P$ is a complex matrix, $A$ is diagonalisable over $mathbb{C}$, and not $mathbb{R}$.
$endgroup$
add a comment |
Your Answer
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%2f3077954%2fdiagonalizability-over-mathbbc-and-mathbbr-respectively%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
All your work seems correct to me, but $A$ is not diagonalisable over $mathbb{R}$, since your eigenvectors aren't over $mathbb{R}$, but over $mathbb{C}$. The diagonalisation would be $PDP^{-1}=A$ where:
$$
P=begin{pmatrix}
1 & -1 & i & -i \
1 & 1 & -1 & -1 \
1 & -1 & -i & i \
1 & 1 & 1 & 1
end{pmatrix},
D=operatorname{diag}(1, -1, i, -i)
$$
And since $P$ is a complex matrix, $A$ is diagonalisable over $mathbb{C}$, and not $mathbb{R}$.
$endgroup$
add a comment |
$begingroup$
All your work seems correct to me, but $A$ is not diagonalisable over $mathbb{R}$, since your eigenvectors aren't over $mathbb{R}$, but over $mathbb{C}$. The diagonalisation would be $PDP^{-1}=A$ where:
$$
P=begin{pmatrix}
1 & -1 & i & -i \
1 & 1 & -1 & -1 \
1 & -1 & -i & i \
1 & 1 & 1 & 1
end{pmatrix},
D=operatorname{diag}(1, -1, i, -i)
$$
And since $P$ is a complex matrix, $A$ is diagonalisable over $mathbb{C}$, and not $mathbb{R}$.
$endgroup$
add a comment |
$begingroup$
All your work seems correct to me, but $A$ is not diagonalisable over $mathbb{R}$, since your eigenvectors aren't over $mathbb{R}$, but over $mathbb{C}$. The diagonalisation would be $PDP^{-1}=A$ where:
$$
P=begin{pmatrix}
1 & -1 & i & -i \
1 & 1 & -1 & -1 \
1 & -1 & -i & i \
1 & 1 & 1 & 1
end{pmatrix},
D=operatorname{diag}(1, -1, i, -i)
$$
And since $P$ is a complex matrix, $A$ is diagonalisable over $mathbb{C}$, and not $mathbb{R}$.
$endgroup$
All your work seems correct to me, but $A$ is not diagonalisable over $mathbb{R}$, since your eigenvectors aren't over $mathbb{R}$, but over $mathbb{C}$. The diagonalisation would be $PDP^{-1}=A$ where:
$$
P=begin{pmatrix}
1 & -1 & i & -i \
1 & 1 & -1 & -1 \
1 & -1 & -i & i \
1 & 1 & 1 & 1
end{pmatrix},
D=operatorname{diag}(1, -1, i, -i)
$$
And since $P$ is a complex matrix, $A$ is diagonalisable over $mathbb{C}$, and not $mathbb{R}$.
answered Jan 18 at 8:31
Yuval GatYuval Gat
9741213
9741213
add a comment |
add a comment |
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%2f3077954%2fdiagonalizability-over-mathbbc-and-mathbbr-respectively%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
2
$begingroup$
$lambda^4 - 1 = (lambda^2 - 1)(lambda^2 + 1) = (lambda - 1)(lambda + 1)(lambda^2 + 1)$; $lambda^4 + 1$ is irreducible over $Bbb R$. Check again! Cheers!
$endgroup$
– Robert Lewis
Jan 18 at 8:26