On Radic's definition for the determinant of a non-square matrices
$begingroup$
Radic gives the following definition for the determinant of non-square matrices
Let $A=(a_{i,j})$ be an $mtimes n$ matrix with $mleq n$. The determinant of $A$, is defined as : $$operatorname{det}(A)=sum_{1leq j_1leqcdotsleq j_mleq n}(-1)^{r+s}operatorname{det}begin{pmatrix} a_{1,j_1}& cdots &a_{1,j_m}\
vdots &ddots&vdots\
a_{m,j_1}&cdots& a_{m,j_m}end{pmatrix}$$
where $j_1,cdots, j_min mathbb N$, $r=1+2+cdots+n$ and $s=j_1+cdots+j_m$. If $m>n$, then we define $operatorname{det}(A)=0$.
My question is, what is the advantage of the last part of the definition, that is "If $m>n$, then we define $operatorname{det}(A)=0$." Why not say if $m>n$ then $operatorname{det}(A):=operatorname{det}(A^T)$? Is there any particular advantage of the way Radic has defined it?
Thank you.
linear-algebra
asked Dec 20 '18 at 4:29
R_DR_D
3,98551533
$endgroup$
add a comment |
$begingroup$
Radic gives the following definition for the determinant of non-square matrices
Let $A=(a_{i,j})$ be an $mtimes n$ matrix with $mleq n$. The determinant of $A$, is defined as : $$operatorname{det}(A)=sum_{1leq j_1leqcdotsleq j_mleq n}(-1)^{r+s}operatorname{det}begin{pmatrix} a_{1,j_1}& cdots &a_{1,j_m}\
vdots &ddots&vdots\
a_{m,j_1}&cdots& a_{m,j_m}end{pmatrix}$$
where $j_1,cdots, j_min mathbb N$, $r=1+2+cdots+n$ and $s=j_1+cdots+j_m$. If $m>n$, then we define $operatorname{det}(A)=0$.
My question is, what is the advantage of the last part of the definition, that is "If $m>n$, then we define $operatorname{det}(A)=0$." Why not say if $m>n$ then $operatorname{det}(A):=operatorname{det}(A^T)$? Is there any particular advantage of the way Radic has defined it?
Thank you.
linear-algebra
asked Dec 20 '18 at 4:29
R_DR_D
3,98551533
$endgroup$
add a comment |
$begingroup$
Radic gives the following definition for the determinant of non-square matrices
Let $A=(a_{i,j})$ be an $mtimes n$ matrix with $mleq n$. The determinant of $A$, is defined as : $$operatorname{det}(A)=sum_{1leq j_1leqcdotsleq j_mleq n}(-1)^{r+s}operatorname{det}begin{pmatrix} a_{1,j_1}& cdots &a_{1,j_m}\
vdots &ddots&vdots\
a_{m,j_1}&cdots& a_{m,j_m}end{pmatrix}$$
where $j_1,cdots, j_min mathbb N$, $r=1+2+cdots+n$ and $s=j_1+cdots+j_m$. If $m>n$, then we define $operatorname{det}(A)=0$.
My question is, what is the advantage of the last part of the definition, that is "If $m>n$, then we define $operatorname{det}(A)=0$." Why not say if $m>n$ then $operatorname{det}(A):=operatorname{det}(A^T)$? Is there any particular advantage of the way Radic has defined it?
Thank you.
linear-algebra
asked Dec 20 '18 at 4:29
R_DR_D
3,98551533
$endgroup$
Radic gives the following definition for the determinant of non-square matrices
Let $A=(a_{i,j})$ be an $mtimes n$ matrix with $mleq n$. The determinant of $A$, is defined as : $$operatorname{det}(A)=sum_{1leq j_1leqcdotsleq j_mleq n}(-1)^{r+s}operatorname{det}begin{pmatrix} a_{1,j_1}& cdots &a_{1,j_m}\
vdots &ddots&vdots\
a_{m,j_1}&cdots& a_{m,j_m}end{pmatrix}$$
where $j_1,cdots, j_min mathbb N$, $r=1+2+cdots+n$ and $s=j_1+cdots+j_m$. If $m>n$, then we define $operatorname{det}(A)=0$.
My question is, what is the advantage of the last part of the definition, that is "If $m>n$, then we define $operatorname{det}(A)=0$." Why not say if $m>n$ then $operatorname{det}(A):=operatorname{det}(A^T)$? Is there any particular advantage of the way Radic has defined it?
Thank you.
linear-algebra
linear-algebra
asked Dec 20 '18 at 4:29
R_DR_D
3,98551533
asked Dec 20 '18 at 4:29
R_DR_D
3,98551533
edited Dec 20 '18 at 13:42
R_D
asked Dec 20 '18 at 4:29
R_DR_D
3,98551533
asked Dec 20 '18 at 4:29
R_DR_D
3,98551533
asked Dec 20 '18 at 4:29
R_DR_D
3,98551533
3,98551533
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%2f3047157%2fon-radics-definition-for-the-determinant-of-a-non-square-matrices%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%2f3047157%2fon-radics-definition-for-the-determinant-of-a-non-square-matrices%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
