This document defines and explains eigen values and eigen vectors. It states that eigen values are scalars associated with linear systems of equations that are used to transform eigenvectors. Eigenvectors are vectors that do not change direction when a linear transformation is applied, but only change by a scalar factor. The document provides properties of eigen values and eigenvectors, explains how to diagonalize matrices, defines quadratic forms and their canonical form, and gives an example of how eigen value analysis is used by oil companies to explore for oil reserves.

1. MATRICS
AND
CALCULUS

2. EIGEN VALUES AND EIGEN VECTORS:
EIGEN VALUES:
Definition:
Eigenvalues are the special set of scalars associated with the
system of linear
Equations.It is mostly used in matrix equations.’Eigen’ is a German word that
means ‘proper’ or ‘characteristic’ .Therefore,the term eigenvalue can be termed as
characteristic value, characteristic root, proper values or latent roots as well . In
simple words, the eigen
Value is a scalar that is used to transform the eigenvector. The basic equation is
AX = X

3. The number or scalar value ““ is an eigenvalue of A
In Mathematics,an eigenvector corresponds to the real non zero eigenvalues
which points in the direction stretched by the transformation whereas
eigenvalue is considered as a factor by which it is stretched. In case,if the
eigenvalue is negative, the direction of thetransformation is negative.
For every real matrix, there is an eigenvalue. Sometimes it might be complex.
The existence of the eigenvalue for the complex matrix is equal to the
fundamental theorem of algebra.

4. EIGEN VECTORS:
Definition:
Eigenvectors are the vectors (non-zero) that do not change the direction when
any linear transformation is applied.It changes by only a scaar factor. In a brief, we can say, if
A is a linear transformation from a vector space V and x is a vector in V, which is not a zero
vector,then v is an eigenvector of A if A(X) is a scalar multiple of x.
An Eigenspace of vector x consists of a set of all eigenvectors with the equivalent eigenvalue
collectively with the zero vector.Though,the zero vector is not an eigenvector.

5. Let us say A is an “n*n” matrix and  is an eigenvalue of matrix A,then x,a non-zero vector, is
called as eigenvector if it satisfies the given below expression;
Ax= x
X is an eigenvector of A corresponding to eigenvalue, .

6. PROPERTIES:
O Sum of eigen values is equal to sum of leading diagonal element.
O Product of eigen values is equal to the determinant of the matrix.
O Eigen vectors are not unique.
O If  is an Eigen values of A, then ^n is a eigen value of A^n.
O Eigen values of a triangular matrix is just its diagonal element.
O Eigen vector of symmetric martices are orthogonal.
O Symmetric Matrices have zero Eigen values.
O Non-symmetric matrices with non repeated eigen values

7. Diagonalization:
A square matrices A is said to be diagonalization if they exit a non-singular
matric be search that D= N^T AN, where N is normalized matric

8. Quadratic Form:
A homogenous polynomial of second degree in any number of variables is called a
Quadratic form.
Canonical form:
A Quadratic form from which contain only the square terms of the variable is said to
be in canonical form.
Q=Y^TDY

9. Converting Quadratic to matric form:

10. Rank,Index,nature and signaturE:
Rank:
Number of Non-zero Eigen value.
Index:
Number of positive Eigen values of A.
Signature:
The different between number of positive and negative Eigen values.

11. Nature:
1) If all Eigen values are positive; then Quadratic form is said to be positive definite.
2) If all Eigen values are negative; then Quadratic form is said to be negative definite.
3) All Eigen values are positive and atleast one zero; then Quadratic form is said to be positive
semi definite.
4) All Eigen values are negative and atleast one zero; then Quadratic form is said to be negative
semi definite.
5) Both positive and negative Eigen values; then Quadratic form is Indefinite.

12. Application:
Eigen value analysis is commonly used by Oil firms to explore
land for Oil. Because Oil, dirt, and other substances all produce linear systems
with varying eigen values, eigen value analysis can help pinpoint where oil
reserves lie. Oil companies set up probes all- around a site to pick up the waves
created by a massive truck vibrating the ground. The waves are modified when
they move through the different substances in the earth. The Oil corporations are
directed to possible drilling sites based on the study of these waves.

13. THANK YOU