# Echelon Matrix

A matrix is said to be echelon marix if :

- Leading entry of each non zero row is 1
- The number of zeros preceeding this entry 1 is larger than number of proceeding rows.

Ex. A =

Consistency of a Algebraic Equations

Note :

An equation is said to be homogeneus if *b* = 0 i.e. of form AX = 0.

A system of equations is said to be consistent, if there are one or more solution for the equation.

A system of equations is said to be inconsistent, if there is no solution. When we reduce a matrix A to echelon form it is called argument matrix A*.

If *p*(A*) = *p*(A) the equations are consistent

If *p*(A*) â‰ *p*(A) the equations are inconsistent.

In a homogeneous system of equation (AX = 0),

If *p*(A) = no. of unknowns (*n*), it has no solution

If *p*(A) < no of unknown (*n*), it has infinite solutions.

Consider an augmented matrix AB which is reduced to echelon form.

If *p*(A) â‰ *p*(AB), system is inconsistent

If *p*(A) = *p*(AB) = no. of unknown (*n*), system is consistent with unique solution.

If *p*(A) = *p*(AB) < number of unknowns (n), system is consistent with infinite solutions.

# Eigen Values and Eigen Vectors

For an *n Ã— n* matrix A and a scalar Î», characteristic matrix of A is

A â€“ Î»I. The equations | A â€“ Î»I | = 0 is called characterisic equation and its roots is called eigen values.

For eigen values (Î») of matrix A, there is a non zero vector X such that AX = Î»X is called characteristic vector (eigen vector) of A.

Relations between Eigen values and Eigen vector

- Î» is a eigen values of a matrix A if and only if there exists a non-zero vector X such that
- AX = Î» X
- If X is a characteristic vector of a matrix A corresponding to the characteristic value Î», then KX is also a characteristic vector of A corresponding to the same characteristic value Î». Here K is any non-zero scalar.
- If X is a eigen vector of a matrix A, then X cannot correspond to more than one eigen value of A.
- Eigen vectors corresponding to distinct characteristic roots of a matrix are linearly independent.

# Properties of Eigen Values

- Sum of eigen values is equal to sum of element of principal diagonal of the matrix i.e.
- Product of eigen values = determinant of the matrix.
- If Î» is eigen value of an orthogonal matrix, A then is also its eigen value.
- If Î» is eigen value of matrix A, then is eigen value of matrix A
^{â€“1}. - If Î»
_{1}, Î»_{2}, Î»_{3}... are eigen values of matrix A, then Î»_{1}, Î»^{n}_{2}, Î»^{n}_{3}... are eigen values of matrix A^{n}.^{n} - Both A and A
^{T}will have same eigen values. - Eigen values of triangular matrix is equal to its diagonal elements.
- Eigen values of real symmetrix matrix are all real.
- Eigen values of skew hermitian matrix are purely imaginary or zero.
- It Î» is eigen value of A, then is eigen value of adj. (A).