Proof of the existence of a skew-symmetric orthogonal matrix with even number of dimensions.
0
How do I prove, that there exists a matrix A with even number of dimensions that is both skew-symmetric and orthogonal:
$$tag{1}A^{T}=-A=A^{-1}.$$
I only found posts, where people just post some matrix and say, that it satisfies both the criteria, but I didn't manage to find a formal proof of the existence.
linear-algebra
asked Dec 2 '18 at 6:35
arstep
32
add a comment |
0
How do I prove, that there exists a matrix A with even number of dimensions that is both skew-symmetric and orthogonal:
$$tag{1}A^{T}=-A=A^{-1}.$$
I only found posts, where people just post some matrix and say, that it satisfies both the criteria, but I didn't manage to find a formal proof of the existence.
linear-algebra
asked Dec 2 '18 at 6:35
arstep
32
The problem formulation seems a bit confused. Perhaps you are trying to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. It should be clear that not every $A$ satisfying (1) must also satisfy (2). But a weak problem formulation would be merely that some (even-dimensioned) matrix satisfies (1), and other $A$ will satisfy (2). However the manner of your presentation seems to want to express a connection between (1) and (2). Do you know what that connection is, or is the discovery of a connection the real crux of your Question?
– hardmath
Dec 2 '18 at 7:04
1
@hardmath Yes, that's right, I want to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. Sorry for the incorrect formulation. I do understand, that not not every A satisfying (1) also satisfies (2), and, since I didn't manage to find posts relevant to my question, I decided to ask it here.
– arstep
Dec 2 '18 at 7:32
Okay, thanks for the clarification. My hint is to think about the real $2times 2$ case. If you get an example there, you can parlay it into all even dimensions.
– hardmath
Dec 2 '18 at 7:37
@hardmath Got it, thanks.
– arstep
Dec 2 '18 at 7:40
add a comment |
0
0
0
How do I prove, that there exists a matrix A with even number of dimensions that is both skew-symmetric and orthogonal:
$$tag{1}A^{T}=-A=A^{-1}.$$
I only found posts, where people just post some matrix and say, that it satisfies both the criteria, but I didn't manage to find a formal proof of the existence.
linear-algebra
asked Dec 2 '18 at 6:35
arstep
32
How do I prove, that there exists a matrix A with even number of dimensions that is both skew-symmetric and orthogonal:
$$tag{1}A^{T}=-A=A^{-1}.$$
I only found posts, where people just post some matrix and say, that it satisfies both the criteria, but I didn't manage to find a formal proof of the existence.
linear-algebra
linear-algebra
asked Dec 2 '18 at 6:35
arstep
32
asked Dec 2 '18 at 6:35
arstep
32
edited Dec 2 '18 at 9:37
asked Dec 2 '18 at 6:35
arstep
32
asked Dec 2 '18 at 6:35
arstep
32
asked Dec 2 '18 at 6:35
arstep
32
32
The problem formulation seems a bit confused. Perhaps you are trying to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. It should be clear that not every $A$ satisfying (1) must also satisfy (2). But a weak problem formulation would be merely that some (even-dimensioned) matrix satisfies (1), and other $A$ will satisfy (2). However the manner of your presentation seems to want to express a connection between (1) and (2). Do you know what that connection is, or is the discovery of a connection the real crux of your Question?
– hardmath
Dec 2 '18 at 7:04
1
@hardmath Yes, that's right, I want to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. Sorry for the incorrect formulation. I do understand, that not not every A satisfying (1) also satisfies (2), and, since I didn't manage to find posts relevant to my question, I decided to ask it here.
– arstep
Dec 2 '18 at 7:32
Okay, thanks for the clarification. My hint is to think about the real $2times 2$ case. If you get an example there, you can parlay it into all even dimensions.
– hardmath
Dec 2 '18 at 7:37
@hardmath Got it, thanks.
– arstep
Dec 2 '18 at 7:40
add a comment |
The problem formulation seems a bit confused. Perhaps you are trying to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. It should be clear that not every $A$ satisfying (1) must also satisfy (2). But a weak problem formulation would be merely that some (even-dimensioned) matrix satisfies (1), and other $A$ will satisfy (2). However the manner of your presentation seems to want to express a connection between (1) and (2). Do you know what that connection is, or is the discovery of a connection the real crux of your Question?
– hardmath
Dec 2 '18 at 7:04
1
@hardmath Yes, that's right, I want to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. Sorry for the incorrect formulation. I do understand, that not not every A satisfying (1) also satisfies (2), and, since I didn't manage to find posts relevant to my question, I decided to ask it here.
– arstep
Dec 2 '18 at 7:32
Okay, thanks for the clarification. My hint is to think about the real $2times 2$ case. If you get an example there, you can parlay it into all even dimensions.
– hardmath
Dec 2 '18 at 7:37
@hardmath Got it, thanks.
– arstep
Dec 2 '18 at 7:40
The problem formulation seems a bit confused. Perhaps you are trying to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. It should be clear that not every $A$ satisfying (1) must also satisfy (2). But a weak problem formulation would be merely that some (even-dimensioned) matrix satisfies (1), and other $A$ will satisfy (2). However the manner of your presentation seems to want to express a connection between (1) and (2). Do you know what that connection is, or is the discovery of a connection the real crux of your Question?
– hardmath
Dec 2 '18 at 7:04
The problem formulation seems a bit confused. Perhaps you are trying to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. It should be clear that not every $A$ satisfying (1) must also satisfy (2). But a weak problem formulation would be merely that some (even-dimensioned) matrix satisfies (1), and other $A$ will satisfy (2). However the manner of your presentation seems to want to express a connection between (1) and (2). Do you know what that connection is, or is the discovery of a connection the real crux of your Question?
– hardmath
Dec 2 '18 at 7:04
1
1
@hardmath Yes, that's right, I want to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. Sorry for the incorrect formulation. I do understand, that not not every A satisfying (1) also satisfies (2), and, since I didn't manage to find posts relevant to my question, I decided to ask it here.
– arstep
Dec 2 '18 at 7:32
@hardmath Yes, that's right, I want to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. Sorry for the incorrect formulation. I do understand, that not not every A satisfying (1) also satisfies (2), and, since I didn't manage to find posts relevant to my question, I decided to ask it here.
– arstep
Dec 2 '18 at 7:32
Okay, thanks for the clarification. My hint is to think about the real $2times 2$ case. If you get an example there, you can parlay it into all even dimensions.
– hardmath
Dec 2 '18 at 7:37
Okay, thanks for the clarification. My hint is to think about the real $2times 2$ case. If you get an example there, you can parlay it into all even dimensions.
– hardmath
Dec 2 '18 at 7:37
@hardmath Got it, thanks.
– arstep
Dec 2 '18 at 7:40
@hardmath Got it, thanks.
– arstep
Dec 2 '18 at 7:40
add a comment |
1 Answer
1
active
oldest
votes
1
Take the example $A=left[begin{array}{cc}0 & I_n\-I_n & 0end{array}right].$
answered Dec 2 '18 at 7:40
John_Wick
1,366111
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
});
}
});
draft saved
draft discarded
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%2f3022316%2fproof-of-the-existence-of-a-skew-symmetric-orthogonal-matrix-with-even-number-of%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
1
Take the example $A=left[begin{array}{cc}0 & I_n\-I_n & 0end{array}right].$
answered Dec 2 '18 at 7:40
John_Wick
1,366111
add a comment |
1
Take the example $A=left[begin{array}{cc}0 & I_n\-I_n & 0end{array}right].$
answered Dec 2 '18 at 7:40
John_Wick
1,366111
add a comment |
1
1
1
Take the example $A=left[begin{array}{cc}0 & I_n\-I_n & 0end{array}right].$
answered Dec 2 '18 at 7:40
John_Wick
1,366111
Take the example $A=left[begin{array}{cc}0 & I_n\-I_n & 0end{array}right].$
answered Dec 2 '18 at 7:40
John_Wick
1,366111
answered Dec 2 '18 at 7:40
John_Wick
1,366111
answered Dec 2 '18 at 7:40
John_Wick
1,366111
answered Dec 2 '18 at 7:40
John_Wick
1,366111
1,366111
add a comment |
add a comment |
draft saved
draft discarded
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
To learn more, see our tips on writing great answers.
draft saved
draft discarded
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%2f3022316%2fproof-of-the-existence-of-a-skew-symmetric-orthogonal-matrix-with-even-number-of%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
The problem formulation seems a bit confused. Perhaps you are trying to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. It should be clear that not every $A$ satisfying (1) must also satisfy (2). But a weak problem formulation would be merely that some (even-dimensioned) matrix satisfies (1), and other $A$ will satisfy (2). However the manner of your presentation seems to want to express a connection between (1) and (2). Do you know what that connection is, or is the discovery of a connection the real crux of your Question?
– hardmath
Dec 2 '18 at 7:04
1
@hardmath Yes, that's right, I want to prove that there exists a matrix (of even dimension) which is both skew-symmetric and orthogonal. Sorry for the incorrect formulation. I do understand, that not not every A satisfying (1) also satisfies (2), and, since I didn't manage to find posts relevant to my question, I decided to ask it here.
– arstep
Dec 2 '18 at 7:32
Okay, thanks for the clarification. My hint is to think about the real $2times 2$ case. If you get an example there, you can parlay it into all even dimensions.
– hardmath
Dec 2 '18 at 7:37
@hardmath Got it, thanks.
– arstep
Dec 2 '18 at 7:40