“Eigenvector”的概念、定义、习题解题方法及翻译-数学辞典

单词

Eigenvector

释义

Eigenvector

A right eigenvector satisfies

(1)

where

is a column Vector. The right Eigenvalues therefore satisfy

(2)

A left eigenvector satisfies

(3)

where

is a row Vector, so

(4)

(5)

where

is the transpose of

.The left Eigenvalues satisfy

(6)

(since

) where

is the Determinantof A. But this is the same equation satisfied by theright Eigenvalues, so the left and right Eigenvalues are the same. Let

be a Matrix formed by the columns of the right eigenvectors and

be a Matrix formed by therows of the left eigenvectors. Let

(7)

Then

(8)

(9)

(10)

But this equation is of the form

where

is a Diagonal Matrix, so it must be truethat

is also diagonal. In particular, if A is a Symmetric Matrix, then theleft and right eigenvectors are transposes of each other. If A is a Self-Adjoint Matrix, then the left andright eigenvectors are conjugate Hermitian Matrices.

Given a Matrix A with eigenvectors , , and and correspondingEigenvalues , , and , then an arbitrary Vector can be written

(11)

Applying the Matrix A,


			(12)

(13)

, it therefore follows that

(14)

so repeated application of the matrix to an arbitrary vector results in a vector proportional to the Eigenvectorhaving the largest Eigenvalue.

See also Eigenfunction, Eigenvalue

References

Arfken, G. ``Eigenvectors, Eigenvalues.'' §4.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 229-237, 1985.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Eigensystems.'' Ch. 11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 449-489, 1992.

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