Is every set of independent eigenvectors of an orthogonally diagonalizable matrix orthogonal?
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Let $f : V rightarrow V$ be a linear operator, where $V$ is a finite dimensional inner product space over $mathbbC$ of dimension $n$.
Suppose we are given that $f$ has $n$ orthogonal eigenvectors $u_1ldots,u_n$, corresponding to the eigenvalues $lambda_1,ldots,lambda_n$.
Now suppose $v_1,ldots,v_n$ is another set of linearly independent eigenvectors of $f$ for the same eigenvalues $lambda_1,ldots,lambda_n$ respectively.
Question: Are these "new" eigenvectors also orthogonal?
If yes, please provide a proof. If not, a counterexample.
To avoid the trivial case, assume that at $lambda_1,ldots,lambda_n$ are NOT pairwise distinct.
linear-algebra eigenvalues-eigenvectors
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up vote
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Let $f : V rightarrow V$ be a linear operator, where $V$ is a finite dimensional inner product space over $mathbbC$ of dimension $n$.
Suppose we are given that $f$ has $n$ orthogonal eigenvectors $u_1ldots,u_n$, corresponding to the eigenvalues $lambda_1,ldots,lambda_n$.
Now suppose $v_1,ldots,v_n$ is another set of linearly independent eigenvectors of $f$ for the same eigenvalues $lambda_1,ldots,lambda_n$ respectively.
Question: Are these "new" eigenvectors also orthogonal?
If yes, please provide a proof. If not, a counterexample.
To avoid the trivial case, assume that at $lambda_1,ldots,lambda_n$ are NOT pairwise distinct.
linear-algebra eigenvalues-eigenvectors
This reads like a verbatim homework assignment. Did you try anything at all?
â Umberto P.
Sep 6 at 11:55
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $f : V rightarrow V$ be a linear operator, where $V$ is a finite dimensional inner product space over $mathbbC$ of dimension $n$.
Suppose we are given that $f$ has $n$ orthogonal eigenvectors $u_1ldots,u_n$, corresponding to the eigenvalues $lambda_1,ldots,lambda_n$.
Now suppose $v_1,ldots,v_n$ is another set of linearly independent eigenvectors of $f$ for the same eigenvalues $lambda_1,ldots,lambda_n$ respectively.
Question: Are these "new" eigenvectors also orthogonal?
If yes, please provide a proof. If not, a counterexample.
To avoid the trivial case, assume that at $lambda_1,ldots,lambda_n$ are NOT pairwise distinct.
linear-algebra eigenvalues-eigenvectors
Let $f : V rightarrow V$ be a linear operator, where $V$ is a finite dimensional inner product space over $mathbbC$ of dimension $n$.
Suppose we are given that $f$ has $n$ orthogonal eigenvectors $u_1ldots,u_n$, corresponding to the eigenvalues $lambda_1,ldots,lambda_n$.
Now suppose $v_1,ldots,v_n$ is another set of linearly independent eigenvectors of $f$ for the same eigenvalues $lambda_1,ldots,lambda_n$ respectively.
Question: Are these "new" eigenvectors also orthogonal?
If yes, please provide a proof. If not, a counterexample.
To avoid the trivial case, assume that at $lambda_1,ldots,lambda_n$ are NOT pairwise distinct.
linear-algebra eigenvalues-eigenvectors
linear-algebra eigenvalues-eigenvectors
asked Sep 6 at 11:48
EpsilonDelta
1107
1107
This reads like a verbatim homework assignment. Did you try anything at all?
â Umberto P.
Sep 6 at 11:55
add a comment |Â
This reads like a verbatim homework assignment. Did you try anything at all?
â Umberto P.
Sep 6 at 11:55
This reads like a verbatim homework assignment. Did you try anything at all?
â Umberto P.
Sep 6 at 11:55
This reads like a verbatim homework assignment. Did you try anything at all?
â Umberto P.
Sep 6 at 11:55
add a comment |Â
1 Answer
1
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votes
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3
down vote
accepted
Consider the $2 times 2$ identity matrix, and the standard basis. Then look at any other basis.
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Consider the $2 times 2$ identity matrix, and the standard basis. Then look at any other basis.
add a comment |Â
up vote
3
down vote
accepted
Consider the $2 times 2$ identity matrix, and the standard basis. Then look at any other basis.
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Consider the $2 times 2$ identity matrix, and the standard basis. Then look at any other basis.
Consider the $2 times 2$ identity matrix, and the standard basis. Then look at any other basis.
answered Sep 6 at 11:51
John Hughes
60k23987
60k23987
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add a comment |Â
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This reads like a verbatim homework assignment. Did you try anything at all?
â Umberto P.
Sep 6 at 11:55