Mixing
Terminology for different types of eigenvalue degeneracy?
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Let $T$ be a linear operator on a finite-dimensional complex vector space $V$, and let $lambda$ be an eigenvalue of $T$ with multiplicity $m$ (defined as the dimension of the subspace of $V$ spanned by generalized eigenvectors of $lambda$).
At least one of the generalized eigenvectors of $V$ must be a “proper” eigenvector, but it may turn out that there are more than one (linearly independent) proper eigenvectors. For example, the operator defined by the matrix
begin{bmatrix}
0 & 0 & 1\
0 & 0 & 1\
0 & 0 & 0
end{bmatrix}
has one eigenvalue, $0$, with multiplicity $3$. But it also seems noteworthy that it has two proper eigenvectors that are linearly independent: $(1,0,0)$ and $(0,1,0)$.
Is there good terminology to express this fact? For example, could I say that the eigenvalue of $0$ for the above operator is “$(2,1)$-degenate,” meaning it has multiplicity $2+1=3$, where there are $2$ eigenvectors and $1$ generalized eigenvector, all of which are linearly independent?
linear-algebra terminology generalized-eigenvector
asked Feb 2 at 23:03
WillGWillG
500310
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add a comment |
$begingroup$
Let $T$ be a linear operator on a finite-dimensional complex vector space $V$, and let $lambda$ be an eigenvalue of $T$ with multiplicity $m$ (defined as the dimension of the subspace of $V$ spanned by generalized eigenvectors of $lambda$).
At least one of the generalized eigenvectors of $V$ must be a “proper” eigenvector, but it may turn out that there are more than one (linearly independent) proper eigenvectors. For example, the operator defined by the matrix
begin{bmatrix}
0 & 0 & 1\
0 & 0 & 1\
0 & 0 & 0
end{bmatrix}
has one eigenvalue, $0$, with multiplicity $3$. But it also seems noteworthy that it has two proper eigenvectors that are linearly independent: $(1,0,0)$ and $(0,1,0)$.
Is there good terminology to express this fact? For example, could I say that the eigenvalue of $0$ for the above operator is “$(2,1)$-degenate,” meaning it has multiplicity $2+1=3$, where there are $2$ eigenvectors and $1$ generalized eigenvector, all of which are linearly independent?
linear-algebra terminology generalized-eigenvector
asked Feb 2 at 23:03
WillGWillG
500310
$endgroup$
$begingroup$
I don't know about a specific terminology for this, but the partition $(2,1)$ in your example is the shape of the Young diagram (or dot diagram) associated to the generalized eigenspace when computing the Jordan canonical form.
$endgroup$
– Christoph
Feb 2 at 23:32
1
$begingroup$
This question introduces "algebraic multiplicity" versus "geometric multiplicity".
$endgroup$
– Roy Simpson
Feb 5 at 19:25
add a comment |
$begingroup$
Let $T$ be a linear operator on a finite-dimensional complex vector space $V$, and let $lambda$ be an eigenvalue of $T$ with multiplicity $m$ (defined as the dimension of the subspace of $V$ spanned by generalized eigenvectors of $lambda$).
At least one of the generalized eigenvectors of $V$ must be a “proper” eigenvector, but it may turn out that there are more than one (linearly independent) proper eigenvectors. For example, the operator defined by the matrix
begin{bmatrix}
0 & 0 & 1\
0 & 0 & 1\
0 & 0 & 0
end{bmatrix}
has one eigenvalue, $0$, with multiplicity $3$. But it also seems noteworthy that it has two proper eigenvectors that are linearly independent: $(1,0,0)$ and $(0,1,0)$.
Is there good terminology to express this fact? For example, could I say that the eigenvalue of $0$ for the above operator is “$(2,1)$-degenate,” meaning it has multiplicity $2+1=3$, where there are $2$ eigenvectors and $1$ generalized eigenvector, all of which are linearly independent?
linear-algebra terminology generalized-eigenvector
asked Feb 2 at 23:03
WillGWillG
500310
$endgroup$
Let $T$ be a linear operator on a finite-dimensional complex vector space $V$, and let $lambda$ be an eigenvalue of $T$ with multiplicity $m$ (defined as the dimension of the subspace of $V$ spanned by generalized eigenvectors of $lambda$).
At least one of the generalized eigenvectors of $V$ must be a “proper” eigenvector, but it may turn out that there are more than one (linearly independent) proper eigenvectors. For example, the operator defined by the matrix
begin{bmatrix}
0 & 0 & 1\
0 & 0 & 1\
0 & 0 & 0
end{bmatrix}
has one eigenvalue, $0$, with multiplicity $3$. But it also seems noteworthy that it has two proper eigenvectors that are linearly independent: $(1,0,0)$ and $(0,1,0)$.
Is there good terminology to express this fact? For example, could I say that the eigenvalue of $0$ for the above operator is “$(2,1)$-degenate,” meaning it has multiplicity $2+1=3$, where there are $2$ eigenvectors and $1$ generalized eigenvector, all of which are linearly independent?
linear-algebra terminology generalized-eigenvector
linear-algebra terminology generalized-eigenvector
asked Feb 2 at 23:03
WillGWillG
500310
asked Feb 2 at 23:03
WillGWillG
500310
asked Feb 2 at 23:03
WillGWillG
500310
asked Feb 2 at 23:03
WillGWillG
500310
asked Feb 2 at 23:03
WillGWillG
500310
500310
$begingroup$
I don't know about a specific terminology for this, but the partition $(2,1)$ in your example is the shape of the Young diagram (or dot diagram) associated to the generalized eigenspace when computing the Jordan canonical form.
$endgroup$
– Christoph
Feb 2 at 23:32
1
$begingroup$
This question introduces "algebraic multiplicity" versus "geometric multiplicity".
$endgroup$
– Roy Simpson
Feb 5 at 19:25
add a comment |
$begingroup$
I don't know about a specific terminology for this, but the partition $(2,1)$ in your example is the shape of the Young diagram (or dot diagram) associated to the generalized eigenspace when computing the Jordan canonical form.
$endgroup$
– Christoph
Feb 2 at 23:32
1
$begingroup$
This question introduces "algebraic multiplicity" versus "geometric multiplicity".
$endgroup$
– Roy Simpson
Feb 5 at 19:25
$begingroup$
I don't know about a specific terminology for this, but the partition $(2,1)$ in your example is the shape of the Young diagram (or dot diagram) associated to the generalized eigenspace when computing the Jordan canonical form.
$endgroup$
– Christoph
Feb 2 at 23:32
$begingroup$
I don't know about a specific terminology for this, but the partition $(2,1)$ in your example is the shape of the Young diagram (or dot diagram) associated to the generalized eigenspace when computing the Jordan canonical form.
$endgroup$
– Christoph
Feb 2 at 23:32
1
1
$begingroup$
This question introduces "algebraic multiplicity" versus "geometric multiplicity".
$endgroup$
– Roy Simpson
Feb 5 at 19:25
$begingroup$
This question introduces "algebraic multiplicity" versus "geometric multiplicity".
$endgroup$
– Roy Simpson
Feb 5 at 19:25
add a comment |
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$begingroup$
I don't know about a specific terminology for this, but the partition $(2,1)$ in your example is the shape of the Young diagram (or dot diagram) associated to the generalized eigenspace when computing the Jordan canonical form.
$endgroup$
– Christoph
Feb 2 at 23:32
1
$begingroup$
This question introduces "algebraic multiplicity" versus "geometric multiplicity".
$endgroup$
– Roy Simpson
Feb 5 at 19:25