Let be a linear transformation represented by a Matrix A. If there is a Vector
such that
(1) |
(2) |
(3) |
(4) |
(5) |
As shown in Cramer's Rule, a system of linear equations has nontrivial solutions only if the Determinant
vanishes, so we obtain the Characteristic Equation
(6) |
(7) |
Assume A has nondegenerate eigenvalues
and corresponding linearly independent
Eigenvectors
which can be denoted
(8) |
(9) |
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
A further remarkable result involving the matrices
and
follows from the definition
(17) |
(18) |
can be found using
(19) |
Assume we know the eigenvalue for
(20) |
(21) |
(22) |
Now consider a Similarity Transformation of A. Let
be the Determinant of A, then
(23) |
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 Separation Theorem, Random Matrix, Schur's Inequalities, Sturmian Separation Theorem, Sylvester's Inertia 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.
© 1996-9 Eric W. Weisstein