is there is relation between nullity and geometric multiplicity
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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
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up vote
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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
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
add a comment |
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?
linear-algebra eigenvalues-eigenvectors
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
linear-algebra eigenvalues-eigenvectors
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
add a comment |
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
add a comment |
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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