An eigenvector of a square matrix Ais a non-zero vector v that, when the matrix is multiplied by v, yields a constant multiple of v, the multiplier being commonly denoted by lambda. That is: Av = lambdav
The number lambda is called the eigenvalue of A corresponding to v.Yes, similar matrices have the same eigenvalues.
Call your matrix A, the eigenvalues are defined as the numbers e for which a nonzero vector v exists such that Av = ev. This is equivalent to requiring (A-eI)v=0 to have a non zero solution v, where I is the identity matrix of the same dimensions as A. A matrix A-eI with this property is called singular and has a zero determinant. The determinant of A-eI is a polynomial in e, which has the eigenvalues of A as roots. Often setting this polynomial to zero and solving for e is the easiest way to compute the eigenvalues of A.
No, in general they do not. They have the same eigenvalues but not the same eigenvectors.
Yes. Simple example: a=(1 i) (-i 1) The eigenvalues of the Hermitean matrix a are 0 and 2 and the corresponding eigenvectors are (i -1) and (i 1). A Hermitean matrix always has real eigenvalues, but it can have complex eigenvectors.
Eigenvalues and eigenvectors are properties of a mathematical matrix.See related Wikipedia link for more details on what they are and some examples of how to use them for analysis.
Yes, similar matrices have the same eigenvalues.
Call your matrix A, the eigenvalues are defined as the numbers e for which a nonzero vector v exists such that Av = ev. This is equivalent to requiring (A-eI)v=0 to have a non zero solution v, where I is the identity matrix of the same dimensions as A. A matrix A-eI with this property is called singular and has a zero determinant. The determinant of A-eI is a polynomial in e, which has the eigenvalues of A as roots. Often setting this polynomial to zero and solving for e is the easiest way to compute the eigenvalues of A.
No, in general they do not. They have the same eigenvalues but not the same eigenvectors.
Yes. Simple example: a=(1 i) (-i 1) The eigenvalues of the Hermitean matrix a are 0 and 2 and the corresponding eigenvectors are (i -1) and (i 1). A Hermitean matrix always has real eigenvalues, but it can have complex eigenvectors.
Carl Sheldon Park has written: 'Real eigenvalues of unsymmetric matrices' -- subject(s): Aeronautics
James V. Burke has written: 'Differential properties of eigenvalues' -- subject(s): Accessible book
Eigenvalues and eigenvectors are properties of a mathematical matrix.See related Wikipedia link for more details on what they are and some examples of how to use them for analysis.
R S. Caswell has written: 'A Fortran code for calculation of Eigenvalues and Eigenfunctions in real potential wells'
Gaetano Fichera has written: 'Numerical and quantitative analysis' -- subject(s): Differential equations, Eigenvalues, Numerical solutions
Hung Chang has written: 'Using parallel banded linear system solvers in generalized Eigenvalue problems' -- subject(s): Eigenvalues
An eigenvector is a vector which, when transformed by a given matrix, is merely multiplied by a scalar constant; its direction isn't changed. An eigenvalue, in this context, is the factor by which the eigenvector is multiplied when transformed.
Gertrude K. Blanch has written: 'Mathieu's equation for complex parameters' -- subject(s): Eigenvalues, Mathieu functions 'On the computation of Mathieu functions'