Solution of Linear Systems of Equations
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Solution of Linear Systems of Equations Consistency Rank Geometric Interpretation. Rank of matrix. Equal to the dimension of the largest square sub-matrix of A that has a non-zero determinant. Example:. has rank 3. Rank of matrix (cont’d).

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Solution of Linear Systems of Equations Consistency Rank Geometric Interpretation

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Solution of linear systems of equations consistency rank geometric interpretation

  • Solution of Linear Systems of Equations

  • Consistency

  • Rank

  • Geometric Interpretation


Rank of matrix

Rank of matrix

Equal to the dimension of the largest square sub-matrix of A that has a non-zero determinant.

Example:

has rank 3


Rank of matrix cont d

Rank of matrix (cont’d)

  • Alternative definition: the maximum number of linearly independent columns (or rows) of A.

Therefore,

rank is not 4 !

Example:


Linear systems in matrix form

Linear Systems in Matrix Form

(1)


Solution of linear systems

Solution of Linear Systems

Each side of the equation

(2)

Can be multiplied by A-1 :

Due to the definition of A-1:

Therefore the solution of (2) is:


Consistency solvability

Consistency (Solvability)

  • A-1 does not exist for every A

  • The linear system of equations A·x=b has a solution, or said to be consistent IFF

    Rank{A}=Rank{A|b}

  • A system is inconsistent when

    Rank{A}<Rank{A|b}

Rank{A} is the maximum number of linearly independent columns or rows of A. Rank can be found by using ERO (Elementary Row Oparations) or ECO (Elementary column operations).


Elementary row and column operations

Elementary row and column operations

  • The following operations applied to the augmented matrix [A|b], yield an equivalent linear system

    • Interchanges: The order of two rows/columns can be changed

    • Scaling: Multiplying a row/column by a nonzero constant

    • Sum: The row can be replaced by the sum of that row and a nonzero multiple of any other row.

One can use ERO and ECO to find the Rank as follows:

EROminimum # of rows with at least one nonzero entry

or

ECOminimum # of columns with at least one nonzero entry


An inconsistent example geometric interpretation

An inconsistent example: Geometric interpretation

ERO:Multiply the first row with

-2 and add to the second row

Rank{A}=1

Rank{A|b}=2 > Rank{A}


Uniqueness of solutions

Uniqueness of solutions

  • The system has a unique solution IFF

    Rank{A}=Rank{A|b}=n

    n is the order of the system

  • Such systems are called full-rank systems


Full rank systems

Full-rank systems

  • If Rank{A}=n

    Det{A}  0  A-1 exists Unique solution


Rank deficient matrices

Rank deficient matrices

  • If Rank{A}=m<n

    Det{A} = 0  A is singular so not invertible

    infinite number of solutions (n-m free variables)

    under-determined system

Rank{A}=Rank{A|b}=1

Consistent so solvable


Ill conditioned system of equations

Ill-conditioned system of equations

  • A small deviation in the entries of A matrix, causes a large deviation in the solution.


Ill conditioned continued

Ill-conditioned continued.....

A linear system of equations is said to be “ill-conditioned” if the coefficient matrix tends to be singular


Gaussian elimination

Back substitution

ERO

Gaussian Elimination

  • By using ERO, matrix A is transformed into an upper triangular matrix (all elements below diagonal 0)

  • Back substitution is used to solve the upper-triangular system


Ill conditioned matrices

Ill-Conditioned Matrices

  • Matrix inversion: Ax = b x = A-1b

    • Assume A is a perfectly known matrix

    • Consider b to be obtained from measurement with some uncertainty

  • Terminology

    • Well-conditioned problem: “small” changes in the data b produce “small” changes in the solution x

    • Ill-conditioned problem: “small” changes in the data b produce “large” changes in the solution x

  • Ill-conditioned matrices

    • Caused by nearly linearly dependent equations

    • Characterized by nearly singular A matrix

    • Solution is not reliable

    • Common problem for large algebraic systems

    • Ill-conditioning quantified by the condition number (covered later)


Ill conditioned matrix example

Ill-Conditioned Matrix Example

  • Example

    • e represents measurement error in b2

    • Two rows (columns) are nearly linearly dependent

  • Analytical solution

  • 10% error (e = 0.1)


Definitions

Definitions

Definition 1: A nonzero vector x is an eigenvector (or characteristic vector) of a square matrix A if there exists a scalar λ such that Ax = λx. Then λ is an eigenvalue (or characteristic value) of A.

Note: The zero vector can not be an eigenvector even though A0 = λ0. But λ = 0 can be an eigenvalue.

Example:


Geometric interpretation of eigenvalues and eigenvectors

Geometric interpretation of Eigenvalues and Eigenvectors

An 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.


Eigenvalues

Eigenvalues

Let x be an eigenvector of the matrix A. Then there must exist an eigenvalue λ such that Ax = λx or, equivalently,

Ax - λx= 0or

(A – λI)x= 0

If we define a new matrix B = A – λI, then

Bx= 0

If B has an inverse then x = B-10 = 0. But an eigenvector cannot be zero.

Thus, it follows that x will be an eigenvector of A if and only if B does not have an inverse, or equivalently det(B)=0, or

det(A – λI)= 0

This is called the characteristic equation of A. Its roots determine the eigenvalues of A.


Eigenvalues examples

Eigenvalues: examples

Example 1: Find the eigenvalues of

two eigenvalues: 1,  2

Note: The roots of the characteristic equation can be repeated. That is, λ1 = λ2 =…= λk. If that happens, the eigenvalue is said to be of multiplicity k.

Example 2: Find the eigenvalues of

λ = 2 is an eigenvector of multiplicity 3.


Eigenvectors

Eigenvectors

To each distinct eigenvalue of a matrix A there will correspond at least one eigenvector which can be found by solving the appropriate set of homogenous equations. If λi is an eigenvalue then the corresponding eigenvector xi is the solution of (A – λiI)xi= 0

Example 1 (cont.):


Eigenvectors1

Eigenvectors

Example 2 (cont.): Find the eigenvectors of

Recall that λ = 2 is an eigenvector of multiplicity 3.

Solve the homogeneous linear system represented by

Let . The eigenvectors of  = 2 are of the form

s and t not both zero.


Properties of eigenvalues and eigenvectors

Properties of Eigenvalues and Eigenvectors

Definition: The trace of a matrix A, designated by tr(A), is the sum of the elements on the main diagonal.

Property 1: The sum of the eigenvalues of a matrix equals the trace of the matrix.

Property 2: A matrix is singular if and only if it has a zero eigenvalue.

Property 3: The eigenvalues of an upper (or lower) triangular matrix are the elements on the main diagonal.

Property 4: If λ is an eigenvalue of A and A is invertible, then 1/λ is an eigenvalue of matrix A-1.


Properties of eigenvalues and eigenvectors1

Properties of Eigenvalues and Eigenvectors

Property 5: If λ is an eigenvalue of A then kλ is an eigenvalue of kA where k is any arbitrary scalar.

Property 6: If λ is an eigenvalue of A then λk is an eigenvalue of Ak for any positive integer k.

Property 8: If λ is an eigenvalue of A then λ is an eigenvalue of AT.

Property 9: The product of the eigenvalues (counting multiplicity) of a matrix equals the determinant of the matrix.


Linearly independent eigenvectors

Linearly independent eigenvectors

Theorem: Eigenvectors corresponding to distinct (that is, different) eigenvalues are linearly independent.

Theorem: If λ is an eigenvalue of multiplicity k of an nn matrix A then the number of linearly independent eigenvectors of A associated with λ is given by m = n - r(A- λI). Furthermore, 1 ≤ m ≤ k.

Example 2 (cont.): The eigenvectors of  = 2 are of the form

s and t not both zero.

 = 2 has two linearly independent eigenvectors


Solution of linear systems of equations consistency rank geometric interpretation

Cayley Hamilton Theorem

Statement : Every square matrix satisfies its characteristic equation.

Possible Questions based on Cayley-Hamilton theorem

1. Find Characteristic Equation of a matrix.

2. Verify Cayley-Hamilton theorem .

3. Find the inverse of a matrix.

4. Find the matrix represented by polynomial of a matrix


Solution of linear systems of equations consistency rank geometric interpretation

Find the characteristic equation of the matrix

Show that the equation is satisfied by A.

Also find A-1

.

Using Cayley - Hamilton theorem, find

If


Solution of linear systems of equations consistency rank geometric interpretation

Some Properties of conjugates and Transpose Conjugates


Solution of linear systems of equations consistency rank geometric interpretation

Thm : Eigen values of a Unitary matrix are of unit modulus

Proof: Let ‘A’ be a Unitary matrix, so .…. (1)

and ‘𝜆’ is eigen value of ‘A’ , then A X = 𝜆 X ……(2)

Taking transposed conjugate in eqn. (2)

…(3)


Solution of linear systems of equations consistency rank geometric interpretation

Cryptography

Cryptography is concerned with keeping communications private.


Solution of linear systems of equations consistency rank geometric interpretation

Cryptography consists of Encryption (encoding) and Decryption (decoding)

Cryptography makes use of a matrix to encode a message.

The receiver of the message decodes it using the inverse of the matrix.

The matrix, used by the sender is called the encoding matrix

and its inverse is called the decoding matrix, which is used by the receiver.


Solution of linear systems of equations consistency rank geometric interpretation

Matrix inverses can provide a simple and effective procedure for encoding and decoding messages. To begin, assign the numbers 1-26 to the letters in the alphabet, as shown below. Also assign the number 0 to a blank to provide for space between words.

Blank A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26


Steps to create a cryptogram

Steps to create a cryptogram

To encode “MY NAME IS RAJ”,

Write the sequence as

13, 25, 0, 14, 1, 13, 5, 0, 9, 19, 0, 18, 1, 10, 0

and the Message matrix can be written as

M =


Steps to encode messages

Steps to ENCODE MESSAGES

To encode a message, choose a 3x3 matrix A that has an inverse and multiply ‘A’ on the left with matrix ‘M’

Let A = , So the encrypted matrix,

X = A . M = .

=


Solution of linear systems of equations consistency rank geometric interpretation

Decoding using Matrix INVERSE

Find the inverse of ‘A’,

Since X = A.M, so M = . X

=

The receiver obtain the sequence of numbers as

13, 25, 0, 14, 1, 13, 5, 0, 9, 19, 0, 18, 1, 10, 0

The message can be retrieved with reference to the table of letters as ‘MY-NAME-IS-RAJ’


Solution of linear systems of equations consistency rank geometric interpretation

Electrical Circuit


Solution of linear systems of equations consistency rank geometric interpretation

In matrix notation,


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