Is the collection of all $2$x$2$ real matrices with both eigen values real dense in $mathbb{M}_2(mathbb{R})$
$begingroup$
Is $M$, the collection of all $2$x$2$ real matrices with both eigen values real dense in $mathbb{M}_2(mathbb{R})$? My answer NO.
Could some one verify my answer and answer the clarification too?
This is my approach. I claim that $bar{M} neq X$. Let A be any (fixed) matrix with a complex eigen value with non-zero imaginary part. I prove that no sequence in $M$ can approach $A$.
Let f : $mathbb{M}_2(mathbb{R})rightarrow mathbb{R}$ be given by $tr(.)^2-4cdot det(.)$. This is clearly a continuous function.
Suppose on the contrary, $exists$ a sequence of matrices $A_n in M$ such that $A_n rightarrow A$ (wrt say, some matrix norm), then $f(A_n) rightarrow f(A)$ (Let |f(A)| = r). That is, a sequence of non-negative real numbers approach a negative number, which is a contradiction because $B_{frac{r}{2}}(f(A))$ contains no point of $f(A_n)$. Thus, $M$ is not dense in $X$.
Clarification:
1) Isn't $bar{M} = M$ itself? (as $bar{M} cannot contain a limit point outside M).
2) Do I need to be more specific about matrix norm for this answer, isn't existence enough?
linear-algebra general-topology
asked Feb 17 '16 at 19:28
user166305user166305
1938
$endgroup$
add a comment |
$begingroup$
Is $M$, the collection of all $2$x$2$ real matrices with both eigen values real dense in $mathbb{M}_2(mathbb{R})$? My answer NO.
Could some one verify my answer and answer the clarification too?
This is my approach. I claim that $bar{M} neq X$. Let A be any (fixed) matrix with a complex eigen value with non-zero imaginary part. I prove that no sequence in $M$ can approach $A$.
Let f : $mathbb{M}_2(mathbb{R})rightarrow mathbb{R}$ be given by $tr(.)^2-4cdot det(.)$. This is clearly a continuous function.
Suppose on the contrary, $exists$ a sequence of matrices $A_n in M$ such that $A_n rightarrow A$ (wrt say, some matrix norm), then $f(A_n) rightarrow f(A)$ (Let |f(A)| = r). That is, a sequence of non-negative real numbers approach a negative number, which is a contradiction because $B_{frac{r}{2}}(f(A))$ contains no point of $f(A_n)$. Thus, $M$ is not dense in $X$.
Clarification:
1) Isn't $bar{M} = M$ itself? (as $bar{M} cannot contain a limit point outside M).
2) Do I need to be more specific about matrix norm for this answer, isn't existence enough?
linear-algebra general-topology
asked Feb 17 '16 at 19:28
user166305user166305
1938
$endgroup$
$begingroup$
Your approach also looks correct.
$endgroup$
– Error 404
Dec 31 '16 at 6:37
add a comment |
$begingroup$
Is $M$, the collection of all $2$x$2$ real matrices with both eigen values real dense in $mathbb{M}_2(mathbb{R})$? My answer NO.
Could some one verify my answer and answer the clarification too?
This is my approach. I claim that $bar{M} neq X$. Let A be any (fixed) matrix with a complex eigen value with non-zero imaginary part. I prove that no sequence in $M$ can approach $A$.
Let f : $mathbb{M}_2(mathbb{R})rightarrow mathbb{R}$ be given by $tr(.)^2-4cdot det(.)$. This is clearly a continuous function.
Suppose on the contrary, $exists$ a sequence of matrices $A_n in M$ such that $A_n rightarrow A$ (wrt say, some matrix norm), then $f(A_n) rightarrow f(A)$ (Let |f(A)| = r). That is, a sequence of non-negative real numbers approach a negative number, which is a contradiction because $B_{frac{r}{2}}(f(A))$ contains no point of $f(A_n)$. Thus, $M$ is not dense in $X$.
Clarification:
1) Isn't $bar{M} = M$ itself? (as $bar{M} cannot contain a limit point outside M).
2) Do I need to be more specific about matrix norm for this answer, isn't existence enough?
linear-algebra general-topology
asked Feb 17 '16 at 19:28
user166305user166305
1938
$endgroup$
Is $M$, the collection of all $2$x$2$ real matrices with both eigen values real dense in $mathbb{M}_2(mathbb{R})$? My answer NO.
Could some one verify my answer and answer the clarification too?
This is my approach. I claim that $bar{M} neq X$. Let A be any (fixed) matrix with a complex eigen value with non-zero imaginary part. I prove that no sequence in $M$ can approach $A$.
Let f : $mathbb{M}_2(mathbb{R})rightarrow mathbb{R}$ be given by $tr(.)^2-4cdot det(.)$. This is clearly a continuous function.
Suppose on the contrary, $exists$ a sequence of matrices $A_n in M$ such that $A_n rightarrow A$ (wrt say, some matrix norm), then $f(A_n) rightarrow f(A)$ (Let |f(A)| = r). That is, a sequence of non-negative real numbers approach a negative number, which is a contradiction because $B_{frac{r}{2}}(f(A))$ contains no point of $f(A_n)$. Thus, $M$ is not dense in $X$.
Clarification:
1) Isn't $bar{M} = M$ itself? (as $bar{M} cannot contain a limit point outside M).
2) Do I need to be more specific about matrix norm for this answer, isn't existence enough?
linear-algebra general-topology
linear-algebra general-topology
asked Feb 17 '16 at 19:28
user166305user166305
1938
asked Feb 17 '16 at 19:28
user166305user166305
1938
asked Feb 17 '16 at 19:28
user166305user166305
1938
asked Feb 17 '16 at 19:28
user166305user166305
1938
asked Feb 17 '16 at 19:28
user166305user166305
1938
1938
$begingroup$
Your approach also looks correct.
$endgroup$
– Error 404
Dec 31 '16 at 6:37
add a comment |
$begingroup$
Your approach also looks correct.
$endgroup$
– Error 404
Dec 31 '16 at 6:37
$begingroup$
Your approach also looks correct.
$endgroup$
– Error 404
Dec 31 '16 at 6:37
$begingroup$
Your approach also looks correct.
$endgroup$
– Error 404
Dec 31 '16 at 6:37
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your $M$ is the pre-image $f^{-1}([0,infty))$, hence closed. So $M$ indeed coincides with its own closure.
The matrix norm does not matter, since they are all equivalent.
answered Feb 17 '16 at 19:35
MooSMooS
27.1k11334
$endgroup$
add a comment |
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%2f1660355%2fis-the-collection-of-all-2x2-real-matrices-with-both-eigen-values-real-dense%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
$begingroup$
Your $M$ is the pre-image $f^{-1}([0,infty))$, hence closed. So $M$ indeed coincides with its own closure.
The matrix norm does not matter, since they are all equivalent.
answered Feb 17 '16 at 19:35
MooSMooS
27.1k11334
$endgroup$
add a comment |
$begingroup$
Your $M$ is the pre-image $f^{-1}([0,infty))$, hence closed. So $M$ indeed coincides with its own closure.
The matrix norm does not matter, since they are all equivalent.
answered Feb 17 '16 at 19:35
MooSMooS
27.1k11334
$endgroup$
add a comment |
$begingroup$
Your $M$ is the pre-image $f^{-1}([0,infty))$, hence closed. So $M$ indeed coincides with its own closure.
The matrix norm does not matter, since they are all equivalent.
answered Feb 17 '16 at 19:35
MooSMooS
27.1k11334
$endgroup$
Your $M$ is the pre-image $f^{-1}([0,infty))$, hence closed. So $M$ indeed coincides with its own closure.
The matrix norm does not matter, since they are all equivalent.
answered Feb 17 '16 at 19:35
MooSMooS
27.1k11334
answered Feb 17 '16 at 19:35
MooSMooS
27.1k11334
answered Feb 17 '16 at 19:35
MooSMooS
27.1k11334
answered Feb 17 '16 at 19:35
MooSMooS
27.1k11334
27.1k11334
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%2f1660355%2fis-the-collection-of-all-2x2-real-matrices-with-both-eigen-values-real-dense%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
$begingroup$
Your approach also looks correct.
$endgroup$
– Error 404
Dec 31 '16 at 6:37