System of Linear Equations

Let F be a field. Consider the n scalars x1, x2, ----- xn in F, which satisfy the conditions

A11 x1 + A12x2 + ----- + A1nxn = y1

A21 x1 + A22x2 + ----- + A2nxn = y2

-

Am1 x1 + Am2x2 + ----- + Amnxn = ym

Where y1, y2,----- ym and Aij, 1 ≤ i ≤ m, 1 ≤ j ≤ n are given elements of F. This system is called a system of m linear equations in n unknowns.

Any n-tuple (x1, x2, -----, xn) of elements of F which satisfies each of the equations is called the solution of the system.

If y1 = y2 = ------ = ym = 0 , we say that system is homogeneous. This system can be represented in matrix form

AX = Y, where

A = ( A11 ⋯ A1n⋮ ⋱ ⋮Am1 ⋯ Amn) , X =[ x1⋮xn] , Y =[ y1⋮yn]

We call A the matrix of coefficient of the system.

Elementary Row Operations

1) The multiplication of any row by a nonzero scalar.2) The replacement of row r by row r plus c times row s, c is any scalar and r ≠

s.3) The interchange of any two rows.

Definition : If A and B are m × n matrices over the field F, we say that B is row-equivalent to A if B can be obtained from A by a finite sequence of elementary row operations.

Row reduced Echelon Matrices

Definition: An m × n matrix R is called a row reduced echelon matrix if,

(a) The first nonzero entry in each nonzero row of R is equal to 1.(b) Each column of R which contains the leading nonzero entry of some row has

all its other entries zero.(c) Every row of R which has all its entries zero occurs below every row which has

a nonzero entry.(d) If rows 1,----, r are the nonzero rows of R and if the leading nonzero entry of

row i occurs in column ki, i = 1, ----- , r, then k1 < k2 < ------- < kr.

The matrix

[ A11 A12 ⋯ A1n y1⋮ ⋱ ⋮

Am1 Am2 ⋯ Amn ym] Is called the augmented matrix of the system AX = Y.

Let R be a row-reduced echelon matrix which is obtained from A by a sequence of elementary row operations. If we perform the same sequence of elementary row operations on the augmented matrix A’ we will arrive at a matrix R’ whose first n columns are the columns of R and whose last column contains certain scalars z1,

------ ,zm. The scalars zi are the entries of the m× 1 matrix Z = [ z1⋮zm] which results

from applying the sequence of row operations to the matrix Y. Then the system AX = Y and RX = Z are equivalent and hence have the same solution.

Theorem: Every nonzero m × n matrix is row equivalent to a unique matrix in row reduced echelon form.

Theorem : Let AX = b and CX = d be two linear systems, each of m equations in n unknowns . If the augmented matrices [A|b] and [C|d] of these systems are row equivalent, then both linear systems have exactly the same solutions.

In the Gauss-Jordan reduction procedure for solving the linear system AX = Y, first transform the augmented matrix of the linear system to row reduced echelon form by using elementary row operations and then find solutions using back substitution.

Exercise: Solve or establish the inconsistency of the following system of equations.

(i) x – 7z = 24x + 3y + 2z = -72x + y – 4z = -1.

(ii) x + y + z + 3 = 0.3x + y – 2z + 2 = 02x + 5y + 7z – 7 = 0.

(iii) 4x – 5y – 2z – 2 = 05x – 4y + 2z + 2 = 0.2x + 2y + 8z – 1 = 0.

Note : A system of linear equations is said to be consistant if there exists a solution for the system. Since the homogeneous system has the trivial solution it is always consistant. There are two possible types of solution to a consistant system. Either the system will have a unique solution or it will have infinite many solutions. If a homogenious system has a unique solution, then, since the trivial solution is always a solution, the trivial solution will be its unique solution. If the system is not homogenious, it is possible that no set of values will satisfy all the equations in the system. If this is the case the system is said to be inconsistent.

In the Gauss – Jordan reduction procedure for solving the linear system AX = b, transform the augmented matrix [A|b] to row reduced echelon form by using elementary row operations.

The linear system that corresponds to the matrix in row reduced echelon form has exactly the same solution as the given linear system.

Result : If a matrix A is invertible the system of linear equations AX = Y has a unique solution given by X = A-1Y.

Eigenvalues and Eigenvectors

Let A = [Aij] be a sequence of matrices of order n and I be a unit matrix of the same order. Then the matrix A –λI, where λ is an indeterminate, is called the characteristic polynomial of A. The equation |A – λI| = 0 is called the characteristic equation of A and its roots are called the characteristic roots or eigenvalues of A.

Now consider the matrix equation, AX = λX, where A = [aij] is a matrix of order n

and X = [ x1⋮xn] is a column vector.

The equation AX = λX represents the set of homogeneous equations

(a11 – λ)x1 + a12x2 + - - - - - + a1nxn = 0

a21x1 + ( a22 - λ)x2 + - - - - - + a2nxn = 0

-

an1x1 + an2 x2+ - - - - - + (amn - λ )xn = 0

This system has a nontrivial solution when |A – λI| = 0., which is the characteristic equation of A. The characteristic polynomial |A – λI| = 0 has n roots, in general. These roots λ1, λ2, - - - - -, λn are known as eigenvalues of A. Corresponding to each eigenvalue, the equation AX = λX has a nonzero vector X, called the eigenvector of A.

Definition : Let X1 , X2 , - - - - - , Xm be n × 1 column vectors. We say that X1 , X2 , - - - - - , Xm are linearly independent if whenever C1X1 + C2 X2 +- - - - - + CmXm = 0 for scalars C1, C2, - - - - , Cm we have C1 = C2 = ------ = Cm = 0 if the vectors are not linearly independent we say that they are linearly dependent.

Exercise : (i) Show that the column vectors [100] ,[010 ] ,[

001] are linearly independent.

(ii) Show that the column vectors [121] ,[010] ,[

101] are linearly dependent.

Exercise : Find eigenvalues and eigenvectors of the following matrices.

(a)[ 6−22−23

−1

2−13 ] (b) [ 3−23

10−35

5−47 ] (c) [200

120

012 ]

Definition: Two matrices A and B are said to be similar if there exists a non-singular matrix P such that P-1AP = B.

Definition : We say that the matrix A is diagonalizable if it is similar to a diagonal matrix. In this case we also say that A can be diagonalized.

Theorem: An n × n matrix A is diagonalizable if and only if it has n linearly independent eigenvectors. In this case, A is similar to a diagonal matrix D, whose diagonal elements are the eigenvalues of A and if D = P-1AP, then P is a matrix whose columns are respectively the n linearly independent eigenvectors of A.