is there is relation between nullity and geometric multiplicity

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












Today while solving some Linear algebra exercises, one question came to my mind



Question: suppose given $n×n$ matrix $A$ has nullity $2$ then what can you say about geometric multiplicity (GM) and algebraic multiplicity (AM) of eigenvalue $0$?



My attempt: Since given that $nullity(A)>0$ that imply $0$ must be an eigenvalue of matrix $A$.



Since nullity of $A$ is dimension of nullspace of $A$ i.e. dimension of solution space of of homogeneous system $Ax=bar0$. Hence, Geometric multiplicity(GM) of eigenvalue $0$ i.e. $GM(0)=nullity(A)=2$? (Is am I right)



and as we know $GM≤ AM$ for every eigenvalue of matrix, hence here we have, $2≤AM(0)$, so $AM(0) ≥2$. Can we say $AM(0)=2$ exactly?










share|cite|improve this question

















  • 2




    Everything is completely correct. The answer to the last question is no, except if $n=2$.
    – amsmath
    Sep 11 at 3:23










  • Thank you @amsmath sir
    – Akash Patalwanshi
    Sep 11 at 3:25














up vote
0
down vote

favorite












Today while solving some Linear algebra exercises, one question came to my mind



Question: suppose given $n×n$ matrix $A$ has nullity $2$ then what can you say about geometric multiplicity (GM) and algebraic multiplicity (AM) of eigenvalue $0$?



My attempt: Since given that $nullity(A)>0$ that imply $0$ must be an eigenvalue of matrix $A$.



Since nullity of $A$ is dimension of nullspace of $A$ i.e. dimension of solution space of of homogeneous system $Ax=bar0$. Hence, Geometric multiplicity(GM) of eigenvalue $0$ i.e. $GM(0)=nullity(A)=2$? (Is am I right)



and as we know $GM≤ AM$ for every eigenvalue of matrix, hence here we have, $2≤AM(0)$, so $AM(0) ≥2$. Can we say $AM(0)=2$ exactly?










share|cite|improve this question

















  • 2




    Everything is completely correct. The answer to the last question is no, except if $n=2$.
    – amsmath
    Sep 11 at 3:23










  • Thank you @amsmath sir
    – Akash Patalwanshi
    Sep 11 at 3:25












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Today while solving some Linear algebra exercises, one question came to my mind



Question: suppose given $n×n$ matrix $A$ has nullity $2$ then what can you say about geometric multiplicity (GM) and algebraic multiplicity (AM) of eigenvalue $0$?



My attempt: Since given that $nullity(A)>0$ that imply $0$ must be an eigenvalue of matrix $A$.



Since nullity of $A$ is dimension of nullspace of $A$ i.e. dimension of solution space of of homogeneous system $Ax=bar0$. Hence, Geometric multiplicity(GM) of eigenvalue $0$ i.e. $GM(0)=nullity(A)=2$? (Is am I right)



and as we know $GM≤ AM$ for every eigenvalue of matrix, hence here we have, $2≤AM(0)$, so $AM(0) ≥2$. Can we say $AM(0)=2$ exactly?










share|cite|improve this question













Today while solving some Linear algebra exercises, one question came to my mind



Question: suppose given $n×n$ matrix $A$ has nullity $2$ then what can you say about geometric multiplicity (GM) and algebraic multiplicity (AM) of eigenvalue $0$?



My attempt: Since given that $nullity(A)>0$ that imply $0$ must be an eigenvalue of matrix $A$.



Since nullity of $A$ is dimension of nullspace of $A$ i.e. dimension of solution space of of homogeneous system $Ax=bar0$. Hence, Geometric multiplicity(GM) of eigenvalue $0$ i.e. $GM(0)=nullity(A)=2$? (Is am I right)



and as we know $GM≤ AM$ for every eigenvalue of matrix, hence here we have, $2≤AM(0)$, so $AM(0) ≥2$. Can we say $AM(0)=2$ exactly?







linear-algebra eigenvalues-eigenvectors






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Sep 11 at 3:17









Akash Patalwanshi

9021716




9021716







  • 2




    Everything is completely correct. The answer to the last question is no, except if $n=2$.
    – amsmath
    Sep 11 at 3:23










  • Thank you @amsmath sir
    – Akash Patalwanshi
    Sep 11 at 3:25












  • 2




    Everything is completely correct. The answer to the last question is no, except if $n=2$.
    – amsmath
    Sep 11 at 3:23










  • Thank you @amsmath sir
    – Akash Patalwanshi
    Sep 11 at 3:25







2




2




Everything is completely correct. The answer to the last question is no, except if $n=2$.
– amsmath
Sep 11 at 3:23




Everything is completely correct. The answer to the last question is no, except if $n=2$.
– amsmath
Sep 11 at 3:23












Thank you @amsmath sir
– Akash Patalwanshi
Sep 11 at 3:25




Thank you @amsmath sir
– Akash Patalwanshi
Sep 11 at 3:25















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',
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


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2912702%2fis-there-is-relation-between-nullity-and-geometric-multiplicity%23new-answer', 'question_page');

);

Post as a guest



































active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes















 

draft saved


draft discarded















































 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2912702%2fis-there-is-relation-between-nullity-and-geometric-multiplicity%23new-answer', 'question_page');

);

Post as a guest













































































這個網誌中的熱門文章

How to combine Bézier curves to a surface?

Mutual Information Always Non-negative

Why am i infinitely getting the same tweet with the Twitter Search API?