Brief Notes On Eigenvector Decomposition

An eigenvalue is one that can be found by using the eigenvectors. In the mathematics of linear algebra, both eigenvalues and eigenvectors are mainly used in the linear transformation analysis. Eigenvalues are one that is always represented in form of a matrix and also with a linear set of equations which is mainly a matrix equation. Eigenvectors are also known as “characteristics roots”.

Definition Of Eigenvalue

The eigenvalue is a special type of scalar associated with linear equation systems. The eigenvalues are represented mainly by matrix form. The word eigen in german means characteristics or proper. The eigenvalue is a scalar representation that is mainly used to transform the eigenvector. The eigenvalue is mainly represented by using the following equation.

AX= λX

The λ used in the equation is a scalar value which is an eigenvalue of A. If the value of the eigenvalue turns out to be negative then the transformation direction is negative. But for every matrix there exist a eigenvalues

Definition Of Eigenvector

The vectors that do not change its direction during linear transformations are known as eigenvectors The value of eigenvectors can only be changed by using the scalar values.

Eigenspace is one that consists of a set of all eigenvectors with all its eigenvalues. If A is a square matrix of order n*n and λ is the eigenvalue of matrix A. Let X be a non zero vector and also an eigenvector then it satisfies the relation.

AX=λX.

Properties Of Eigenvalues

1. Eigenvalues of real symmetric matrices are always real.
2. The Eigenvalues of skew-symmetric matrices are either imaginary or zero.
3. The eigenvalues of the unit and orthogonal matrices are always  | λ|=1.
4. Properties of eigenvalues are always equal to | λ|=1.
5. If A and B are two matrices with the same order(in rows and columns) then the eigenvalue of BA=eigenvalues of AB.
6. The Sum of eigenvalues of A is always equal to the trace(A).

Properties Of Eigenvectors

1. Eigenvectors have distinct eigenvalues. This type of eigenvectors are always linearly independent.
2. The zero matrix or singular matrix always has zero eigenvalues.
3. If A is a square matrix the λ=0 cannot exist for that matrix.
4. If λ is an eigenvalue and A is a square matrix then kλ is an eigenvalue of kA.
5. If A is a square matrix of order n*n and λ is an eigenvalue of A. If n>=0 is an integer then we can find that λ^n is also an eigenvalue of  A^n.
6. If A is a square matrix of order n*n and λ is an eigenvalue of A. If p(x) is a polynomial then p(λ) is an eigenvalue in the matrix of p(A).
7. If A is a square matrix of order n*n and λ is an eigenvalue of A, then  λ-1 is always equal to A-1.
8. If A is a square matrix of order n*n and λ is an eigenvalue of A, then λ is an eigenvalue of the transpose of A.

Eigenvalue Decomposition

It is a  process of matrix factorisation into its canonical form. The matrices that are diagonalizable can be factored into its canonical form.

If A is a square matrix of the order of n*n then the matrix A can be diagonalized by using the formula

D=B^-1AB.

Where D is a diagonalisation matrix. The B is the one that contains eigenvectors of all eigenvalues in the form of a matrix.

Single Value Decomposition

The single value decomposition (SVD) is a process that contains the U and V matrix in the same order as the matrix of A. The matrix U is always a normal matrix that contains the eigenvalues of A*AT. The V matrix is always a transpose of the multiplication of AT*A. The Σ is an on which contains the root of eigenvalues. Thus single value decomposition can be represented by using the formula

A=U*Σ*VT

Signature In Eigenvalues Decomposition

Signature = number of positive terms – number of negative terms

S=2p-r

The number of positive terms in the canonical form is an index(p).

Nature of roots based on the sign of Eigenvalue :

Positive Definite

The nature of the matrix is said to be positive definite if all of its eigenvalues are positive.

Eg: eigenvalues= 1,3,6

Negative definite

The nature of the matrix is said to be negative definite if all of its eigenvalues are negative.

Eg: eigenvalues=-1,-3,-6.

Positive semidefinite

The nature of the matrix is said to be positive semi-definite if all of its eigenvalues are positive and one of its eigenvalues is zero.

Eg:eigenvalue= 1,0,5

The nature of the matrix is said to be positive definite if all of its eigenvalues are positive.

Negative semidefinite

The nature of the matrix is said to be negative definite if all of its eigenvalues are negative and one of its eigenvalues is zero.

Eg: eigenvalue: -1,0,-3

Indefinite

The nature of a matrix is said to be indefinite if it contains a mixture of all negative and positive eigenvalues.

Eg; 1,-1,-9.

Conclusion

Thus in single value decomposition and eigenvalue decomposition, the diagonalization of matrix and orthogonal transformations stands as an important step.The decomposition process of eigenvalues is also helpful in finding the nature of roots of the given matrix.