الرحيم الرحمن الله بسم

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EIGENVALUES

DefinitionEigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values, proper values, or latent roots.

The eigenvalue for a given factor measures the variance in all the variables which is accounted for by that factor. The ratio of eigenvalues is the ratio of explanatory importance of the factors with respect to the variables. If a factor has a low eigenvalue, then it is contributing little to the explanation of variances in the variables and may be ignored as redundant with more important factors. Eigenvalues measure the amount of variation in the total sample accounted for by each factor.

Eigenvalues •To select how many factors to use, consider eigenvalues from a principal components analysis •Two interpretations: –eigenvalue equivalent number of variables ≅which the factor represents –eigenvalue amount of variance in the data described by the factor. ≅•Rules to go by: –number of eigenvalues> 1 –

History

• In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes.

• At the start of the 20th century, Hilbert studied the eigenvalues of integral

operators by viewing the operators as infinite matrices. He was the first to use

the German word Eigen, which means "own", to denote eigenvalues and

eigenvectors in 1904,though he may have been following a related usage by

Helmholtz. For some time, the standard term in English was "proper value", but

the more distinctive term "eigenvalue" is standard today.

Uses

The determination of the eigenvalues and eigenvectors of a system is

extremely important in physics and engineering, where it is equivalent

to matrix diagonalization and arises in such common applications as

stability analysis, the physics of rotating bodies, and small oscillations of

vibrating systems, to name only a few.

Also used in the development of computer programs and the graphic design and other objects such as the objects developed in the 3D Studio Max, Adobe Photoshop, Primavera, Flash, AutoCAD, Dreamweaver etc.

Uses Cont.

Eigenvector

In linear algebra, an eigenvector or characteristic vector of a square matrix is a vector that does not change its direction under the associated linear transformation. In other words—if v is a vector that is not zero, then it is an eigenvector of a square matrix A if Av is a scalar multiple of v.

Each eigenvalue is paired with a corresponding so-called eigenvector (or, in general, a corresponding right eigenvector and a corresponding left eigenvector; there is no analogous distinction between left and right for eigenvalues).

In this shear mapping the red arrow changes direction but the blue arrow does not.

The blue arrow is an eigenvector of this shear mapping because it doesn't change

direction, and since its length is unchanged, its eigenvalue is 1.

• In general, a matrix acts on a vector by changing both its magnitude and its direction. However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged (or possibly reversing it). These vectors are the eigenvectors of the matrix.

• A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negative if its direction is reversed. This factor is the eigenvalue associated with that eigenvector.

Properties of Eigenvalues and Eigenvectors

• Property 1: The sum of the eigenvalues of a matrix equals the trace of the matrix.

• Property 2: A matrix is singular if and only if it has a zero eigenvalue.

• Property 3: The eigenvalues of an upper (or lower) triangular matrix are the elements on the main diagonal.

• Property 4: If λ is an eigenvalue of A and A is invertible, then 1/λ is an eigenvalue of matrix A-1.

• Property 5: If λ is an eigenvalue of A then kλ is an eigenvalue of kA where k is any arbitrary scalar.

• Property 6: If λ is an eigenvalue of A then λk is an eigenvalue of Ak for any positive integer k.

• Property 8: If λ is an eigenvalue of A then λ is an eigenvalue of AT.

• Property 9: The product of the eigenvalues (counting multiplicity) of a matrix equals the determinant of the matrix.

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