“Eigenvalue”的概念、定义、习题解题方法及翻译-数学辞典

单词

Eigenvalue

释义

Eigenvalue

Let be a linear transformation represented by a Matrix A. If there is a Vector such that

(1)

for some Scalar

, then

is the eigenvalue of A with corresponding (right) Eigenvector

. Letting A be a

Matrix,

(2)

with eigenvalue

, then the corresponding Eigenvectors satisfy

(3)

which is equivalent to the homogeneous system

(4)

Equation (4) can be written compactly as

(5)

where I is the Identity Matrix.

As shown in Cramer's Rule, a system of linear equations has nontrivial solutions only if the Determinantvanishes, so we obtain the Characteristic Equation

(6)

If all

s are different, then plugging these back in gives

independent equations for the

componentsof each corresponding Eigenvector. The Eigenvectors will then be orthogonal and the systemis said to be nondegenerate. If the eigenvalues are

-fold Degenerate, then the system is said to be degenerate andthe Eigenvectors are not linearly independent. In such cases, the additional constraint that theEigenvectors be orthogonal,

(7)

where

is the Kronecker Delta, can be applied to yield

additional constraints, thus allowing solutionfor the Eigenvectors.

Assume A has nondegenerate eigenvalues and corresponding linearly independentEigenvectors which can be denoted

(8)

Define the matrices composed of eigenvectors

(9)

and eigenvalues

(10)

where

is a Diagonal Matrix. Then






			(11)

(12)

Furthermore,


			(13)

By induction, it follows that for

(14)

The inverse of A is

(15)

where the inverse of the Diagonal Matrix D is trivially given by

(16)

Equation (14) therefore holds for both Positive and Negative

A further remarkable result involving the matrices and follows from the definition


			(17)

Since D is a Diagonal Matrix,



			(18)

can be found using

(19)

Assume we know the eigenvalue for

(20)

Adding a constant times the Identity Matrix to A,

(21)

so the new eigenvalues equal the old plus

. Multiplying A by a constant

(22)

so the new eigenvalues are the old multiplied by

Now consider a Similarity Transformation of A. Let be the Determinant of A, then


			(23)

so the eigenvalues are the same as for A.

See also Brauer's Theorem,Condition Number, Eigenfunction, Eigenvector, Frobenius Theorem,Gersgorin Circle Theorem, Lyapunov's First Theorem, Lyapunov's Second Theorem,Ostrowski's Theorem, Perron's Theorem, Perron-Frobenius Theorem, Poincaré SeparationTheorem, Random Matrix,Schur's Inequalities, Sturmian Separation Theorem, Sylvester'sInertia Law, Wielandt's Theorem

References

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

Nash, J. C. ``The Algebraic Eigenvalue Problem.'' Ch. 9 in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 102-118, 1990.

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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