How to understand the convergence to the dominant eigenvalue-eigenvector pair from the Power Method?
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For the Power Method I make an initial guess for the dominant eigenvector of matrix A. It can be expressed as a linear combination of the set of eigenvectors of A.
Why is it that the Power Method converges to the dominant eigenvector? Is it that the initial guess contains a non-zero component along the dominant eigenvector (by assumption) and since it has the largest (by magnitude) eigenvalue of A the scaling within the recurrence relation "brings" us closer and closer to the dominant eigenvector while the non-dominant eigenvectors are being "silenced" by the scaling?
linear-algebra eigenvalues-eigenvectors algorithms
asked Dec 21 '18 at 3:48
G. LCG. LC
1033
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add a comment |
$begingroup$
For the Power Method I make an initial guess for the dominant eigenvector of matrix A. It can be expressed as a linear combination of the set of eigenvectors of A.
Why is it that the Power Method converges to the dominant eigenvector? Is it that the initial guess contains a non-zero component along the dominant eigenvector (by assumption) and since it has the largest (by magnitude) eigenvalue of A the scaling within the recurrence relation "brings" us closer and closer to the dominant eigenvector while the non-dominant eigenvectors are being "silenced" by the scaling?
linear-algebra eigenvalues-eigenvectors algorithms
asked Dec 21 '18 at 3:48
G. LCG. LC
1033
$endgroup$
$begingroup$
The power method does not always converge. Some conditions are needed to guarantee convergence. One sufficient condition is that the dominant eigenvalue is simple and the initial vector does not lie in the sum of the the generalised eigenspaces for other eigenvalues. This is always satisfied if, for instance, the matrix is nonnegative and irreducible and the initial vector is nonnegative but nonzero.
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– user1551
Dec 21 '18 at 9:54
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@user1551 That's fine, however, I am just checking my understanding of convergence toward the dominant eigenpair with the community. Other than that thanks for the side note.
$endgroup$
– G. LC
Dec 21 '18 at 13:22
add a comment |
$begingroup$
For the Power Method I make an initial guess for the dominant eigenvector of matrix A. It can be expressed as a linear combination of the set of eigenvectors of A.
Why is it that the Power Method converges to the dominant eigenvector? Is it that the initial guess contains a non-zero component along the dominant eigenvector (by assumption) and since it has the largest (by magnitude) eigenvalue of A the scaling within the recurrence relation "brings" us closer and closer to the dominant eigenvector while the non-dominant eigenvectors are being "silenced" by the scaling?
linear-algebra eigenvalues-eigenvectors algorithms
asked Dec 21 '18 at 3:48
G. LCG. LC
1033
$endgroup$
For the Power Method I make an initial guess for the dominant eigenvector of matrix A. It can be expressed as a linear combination of the set of eigenvectors of A.
Why is it that the Power Method converges to the dominant eigenvector? Is it that the initial guess contains a non-zero component along the dominant eigenvector (by assumption) and since it has the largest (by magnitude) eigenvalue of A the scaling within the recurrence relation "brings" us closer and closer to the dominant eigenvector while the non-dominant eigenvectors are being "silenced" by the scaling?
linear-algebra eigenvalues-eigenvectors algorithms
linear-algebra eigenvalues-eigenvectors algorithms
asked Dec 21 '18 at 3:48
G. LCG. LC
1033
asked Dec 21 '18 at 3:48
G. LCG. LC
1033
asked Dec 21 '18 at 3:48
G. LCG. LC
1033
asked Dec 21 '18 at 3:48
G. LCG. LC
1033
asked Dec 21 '18 at 3:48
G. LCG. LC
1033
1033
$begingroup$
The power method does not always converge. Some conditions are needed to guarantee convergence. One sufficient condition is that the dominant eigenvalue is simple and the initial vector does not lie in the sum of the the generalised eigenspaces for other eigenvalues. This is always satisfied if, for instance, the matrix is nonnegative and irreducible and the initial vector is nonnegative but nonzero.
$endgroup$
– user1551
Dec 21 '18 at 9:54
$begingroup$
@user1551 That's fine, however, I am just checking my understanding of convergence toward the dominant eigenpair with the community. Other than that thanks for the side note.
$endgroup$
– G. LC
Dec 21 '18 at 13:22
add a comment |
$begingroup$
The power method does not always converge. Some conditions are needed to guarantee convergence. One sufficient condition is that the dominant eigenvalue is simple and the initial vector does not lie in the sum of the the generalised eigenspaces for other eigenvalues. This is always satisfied if, for instance, the matrix is nonnegative and irreducible and the initial vector is nonnegative but nonzero.
$endgroup$
– user1551
Dec 21 '18 at 9:54
$begingroup$
@user1551 That's fine, however, I am just checking my understanding of convergence toward the dominant eigenpair with the community. Other than that thanks for the side note.
$endgroup$
– G. LC
Dec 21 '18 at 13:22
$begingroup$
The power method does not always converge. Some conditions are needed to guarantee convergence. One sufficient condition is that the dominant eigenvalue is simple and the initial vector does not lie in the sum of the the generalised eigenspaces for other eigenvalues. This is always satisfied if, for instance, the matrix is nonnegative and irreducible and the initial vector is nonnegative but nonzero.
$endgroup$
– user1551
Dec 21 '18 at 9:54
$begingroup$
The power method does not always converge. Some conditions are needed to guarantee convergence. One sufficient condition is that the dominant eigenvalue is simple and the initial vector does not lie in the sum of the the generalised eigenspaces for other eigenvalues. This is always satisfied if, for instance, the matrix is nonnegative and irreducible and the initial vector is nonnegative but nonzero.
$endgroup$
– user1551
Dec 21 '18 at 9:54
$begingroup$
@user1551 That's fine, however, I am just checking my understanding of convergence toward the dominant eigenpair with the community. Other than that thanks for the side note.
$endgroup$
– G. LC
Dec 21 '18 at 13:22
$begingroup$
@user1551 That's fine, however, I am just checking my understanding of convergence toward the dominant eigenpair with the community. Other than that thanks for the side note.
$endgroup$
– G. LC
Dec 21 '18 at 13:22
add a comment |
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$begingroup$
The power method does not always converge. Some conditions are needed to guarantee convergence. One sufficient condition is that the dominant eigenvalue is simple and the initial vector does not lie in the sum of the the generalised eigenspaces for other eigenvalues. This is always satisfied if, for instance, the matrix is nonnegative and irreducible and the initial vector is nonnegative but nonzero.
$endgroup$
– user1551
Dec 21 '18 at 9:54
$begingroup$
@user1551 That's fine, however, I am just checking my understanding of convergence toward the dominant eigenpair with the community. Other than that thanks for the side note.
$endgroup$
– G. LC
Dec 21 '18 at 13:22