Vector Calculus & Linear Algebra

EIGEN VALUES AND EIGEN VECTORS

By: Taher K D

IntroductionIn linear algebra, an eigenvector or characteristic vector of

a square matrix is a vector that points in a direction which is

invariant under the associated linear transformation. In other

words, if v is a vector which is not zero, then it is an

eigenvector of a square matrix A if Av is a scalar multiple of v.

This condition could be written as the equation

Av = λv

where λ is a number (also called a scalar) known as the

eigenvalue or characteristic value associated with the

eigenvector v.

Geometric interpretation of eigenvalues and eigenvectors

A n×n matrix A multiplied by n×1 vector x results in another n×1

vector y=Ax. Thus A can be considered as a transformation

matrix.

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.

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.

Two dimensional example

Consider the transformation matrix A, given by,

The eigenvectors v of this transformation satisfy the equation,

Av = λv

Rearrange this equation to obtain

(A-λI)v=0

which has a solution only when its determinant| A − λI | equals zero. Set the determinant to zero to obtain the polynomial equation,

known as the characteristic polynomial of the matrix A. In this case, it has the roots λ = 1 and λ = 3.

For λ = 1, the equation becomes,

which has the solution,

For λ = 3, the equation becomes,

which has the solution,

Thus, the vectors v and w are eigenvectors of A associated with the eigenvalues λ = 1 and λ = 3, respectively.

The figure shows the effect of this transformation on point coordinates in the plane.

The transformation matrix A = preserves the direction of vectors parallel to v = (1,−1)T (in purple) and w = (1,1)T (in blue). The vectors in

red are not parallel to either eigenvector, so, their directions are changed by the transformation. See also: An extended version, showing

all four quadrants.

Three dimensional example

The eigenvectors v of the 3×3 matrix A,

satisfy the equation

This equation has solutions only if the determinant | A − λI | equals zero, which yields the characteristic polynomial,

with the roots λ = 1, λ = 2 and λ = 3.

Associated with the roots λ = 1, λ = 2 and λ = 3 are the respective eigenvectors,

Diagonal matricesMatrices with entries only along the main diagonal are called diagonal matrices. It is easy to see that the eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix A,

The characteristic polynomial of A is given by

which has the roots λ = 1, λ = 2 and λ = 3.

Associated with these roots are the eigenvectors,

respectively.

Triangular matrices

The eigenvalues of triangular matrices are the elements of the main diagonal, in the same way as for diagonal matrices. Consider the lower triangular matrix A,

The characteristic polynomial of A is given by

which has the roots λ = 1, λ = 2 and λ = 3. Associated with these roots are the eigenvectors,

respectively.

PropertiesLet A be an arbitrary matrix of complex

numbers with eigenvalues λ1 λ2…λn , Then

The trace of A defined as the sum of its diagonal

elements, Is also the sum of all eigenvalues:

The determinant of A is the product of all eigenvalues:

If A is invertible, then the eigenvalues of A-1 are

. Clearly, the geometric multiplicities coincide.

Moreover, since the characteristic polynomial of the

inverse is the reciprocal polynomial for that of the

original, they share the same algebraic multiplicity.

The eigenvalues kth of the power of A i.e. the

eigenvalues of Ak for any positive integer K are

The matrix A is invertible if and only if all the

eigenvalues λi are nonzero.

If A is Hermitian, then every eigenvalue is real. The

same is true of any a symmetric real matrix. If A is also

positive-definite, positive-semi definite, negative-definite,

or negative-semi definite every eigenvalue is positive,

non-negative, negative, or non-positive respectively.

If λ is an eigenvalue of A then kλ is an eigenvalue of kA

where k is any arbitrary scalar.

If λ is an eigenvalue of A then λ is an eigenvalue of AT.

Application of Eigen values and Eigen vectors

Scaling

Matrix :

Characteristic polynomial :

Eigen Values :

Algebric multiplicity :

Geometric multiplicity :

Eigenvectors : All non-zero vectors

Unequal Scaling

Matrix :

Characteristic polynomial :

Eigen Values :

Algebric multiplicity :

Geometric multiplicity :

Eigenvectors :

Rotation

Matrix :

Characteristic polynomial :

Eigen Values :

Algebric multiplicity :

Geometric multiplicity :

Eigenvectors :

Hyperbolic Rotation

Matrix :

Characteristic polynomial :

Eigen Values :

Algebric multiplicity :

Geometric multiplicity :

Eigenvectors :

Horizontal Shear

Matrix :

Characteristic polynomial :

Eigen Values :

Algebric multiplicity :

Geometric multiplicity :

Eigenvectors :

Thank You