MOHAMMAD IMRANDEPARTMENT OF APPLIED SCIENCES

JAHANGIRABAD EDUCATIONAL GROUP OF INSTITUTESwww.jit.edu.in

Matrix Mathematics

• Matrices are very useful in engineering calculations. For example, matrices are used to:– Efficiently store a large number of values (as we have

done with arrays in MATLAB)– Solve systems of linear simultaneous equations– Transform quantities from one coordinate system to

another

• Several mathematical operations involving matrices are important

Outline Basics:

Operations on matrices Transpose of the matrices Types of matrices Determinant of matrix Linear systems of algebraic equations

Matrix rank, existence of a solutionInverse of a matrixNormal form of the matrixRank of matrix by using the normal form Non-singular matrices P & Q which makes normal form with

given matrix A as PAQ

Outline cont’

Consistency Eigen values and Eigenvectors

Review: Properties of Matrices

• A matrix is a one-or two dimensional array• A quantity is usually designated as a matrix by bold face

type: A• The elements of a matrix are shown in square brackets:

• The dimension (size) of a matrix is defined by the number of rows and number of columns

• Examples:

3 × 3: 2×4:

Review: Properties of Matrices cont.

• An element of a matrix is usually written in lower case, with its row number and column number as subscripts :

Review: Properties of Matrices cont.

• Matrix Addition• Multiplication of a Matrix by a Scalar• Matrix Multiplication• Matrix Transposition • Finding the Determinate of a Matrix• Matrix Inversion

Matrix Operations

• Matrix must be the same size in order to add

• Matrix addition is commutative:

A + B = B + A

Matrix Addition

Multiplication of a Matrix by a Scalar

• To multiple a matrix by a scalar, multiply each element by the scalar:

• We often use this fact to simplify the display of matrices with very large (or very small) values:

Multiplication of Matrices

To multiple two matrices together, the matrices must have compatible sizes:

This multiplication is possible only if the number of columns in A is the same as the number of rows in B

The resultant matrix C will have the same number of rows as A and the same number of columns

as B

Multiplication of Matrices

• Consider these matrices:

• Can we find this product?

• What will be the size of C?

Yes, 3 columns of A = 3 rows of B

2 X 2: 2 rows in A, 2 columns in B

Multiplication of Matrices

• Element ij of the product matrix is computed by multiplying each element of row i of the first matrix by the corresponding element of column j of the second matrix, and summing the results

• This is best illustrated by example

Example – Matrix Multiplication

Find

We know that matrix C will be 2 × 2 Element c11 is found by multiplying terms of row 1

of A and column 1 of B:

Example – Matrix Multiplication

• Element c12 is found by multiplying terms of row 1 of A and column 2 of B:

Example – Matrix Multiplication

• Element c21 is found by multiplying terms of row 2 of A and column 1 of B:

Example – Matrix Multiplication

• Element c22 is found by multiplying terms of row 2 of A and column 2 of B:

Example – Matrix Multiplication

• Solution:

Matrix Multiplication

• In general, matrix multiplication is not commutative:

AB ≠ BA

Transpose of a Matrix

• The transpose of a matrix by switching its row and columns

• The transpose of a matrix is designated by a superscript T:

Types of Matrices 1. Row Matrix : A matrix which has only one row and n

numbers of columns called “Row Matrix”.Ex : - [ 3 4 6 7 8 ………………n]

2. Column Matrix : A Matrix which has only one column and n numbers of rows called “column Matrix”.

3567....n

Square Matrix : A matrix which has equal number of rows and columns called “Square Matrix”.

Where m =n i.e the number of rows and columns are equal

Types of Matrices

Diagonal Matrix : Diagonal matrix is a matrix in which all elements are zero except the diagonal elements.

Remark : Diagonal matrix is a type of square matrix.

Types of Matrices

Scalar Matrix : It is a type of square matrix but

its all diagonal elements are exactly similar and remaining elements should be zero

Where m = n, i.e the number of rows and columns are equal

Types of Matrices

Unit matrix : A Diagonal matrix which has all

its diagonal elements as 1 called “Unit Matrix”

Remark : Except diagonal elements all elements should be zero.

Types of Matrices Types of Matrices

Null Matrix : A matrix whose all elements are zero

called “Null Matrix”.

Remark: This matrix is also type of square matrix.

Types of Matrices

Symmetric Matrix :A matrix which is equal to its transpose

said to be “Symmetric Matrix”

A =

We can see that A =AT

Types of Matrices

Skew - Symmetric Matrix : A matrix which is equal

to its negative of its transpose said to be “Skew-Symmetric Matrix”

A =

We can see that A = - AT

Types of Matrices Types of Matrices

Lower Triangular matrix :- If all the elements below the diagonal are

zero then this type of matrix is called “Lower Triangular matrix”

For Ex.

Types of Matrices

Types of Matrices

Upper Triangular matrix :- if all the elements above the diagonal are

zero then this type of matrix is called “Upper triangular matrix”

For Ex.

Identity Matrix (Unit Matrix):- A matrix is said to be identity

matrix if all the diagonal elements are 1 and remaining elements should be zero.

Types of Matrices

Equal Matrices :- Those matrices which has equal number

of rows as well column and all elements should be same said to be “Equal Matrix”.

and are equal matrices

Types of Matrices

Equivalence Matrix :-Those matrices which has

equal number of rows as well column but not all elements are same said to be “Equivalence Matrix”.

and

Types of Matrices

Orthogonal matrix :-An orthogonal matrix is one

whose transpose is also its inverse. AT = A-1

Types of Matrices

Determinate of a Matrix

• The determinate of a square matrix is a scalar quantity that has some uses in matrix algebra. Finding the determinate of 2 × 2 and 3 × 3 matrices can be done relatively easily:

• The determinate is designated as |A| or det(A) of 2 ×2:

Determinate of a Matrix

• 3 × 3:

Matrix Rank

The rank of a matrix is simply the number of independent row vectors in that matrix.

or The number of non-zero rows in the matrix. The transpose of a matrix has the same rank as the

original matrix. To find the rank of a matrix by hand, use Gauss

elimination and the linearly dependant row vectors will fall out, leaving only the linearly independent vectors, the number of which is the rank.

Matrix inverse

The inverse of the matrix A is denoted as A-1

By definition, AA-1 = A-1A = I, where I is the identity matrix.

Theorem: The inverse of an nxn matrix A exists if and only if the rank A = n.

Gauss-Jordan elimination can be used to find the inverse of a matrix by hand.

Inverse of a 2 x 2 matrix Procedure

There is a simple procedure to find the inverse of a two by two matrix. This procedure only works for the 2 x 2 case. Find the inverse of

∆= delta= difference of product of diagonal elements

Determine whether or not the inverse actually exists. We will define

∆ =

In order for the inverse of a 2 x 2 matrix to exist, ∆ cannot equal to zero.

If happens ∆ to be zero, then we conclude the inverse does not exist and we stop all calculations.

In our case ∆ = 1, so we can proceed.

As (2)2-1(3);

∆ is the difference of the product of the diagonal elements of the matrix.

Inverse of a 2 x 2 matrix Procedure

Inverse of a 2 x 2 matrix

Step 2. Reverse the entries of the main diagonal consisting of the

two 2’s. In this case, no apparent change is noticed. Step 3. Reverse the signs of the other diagonal

entries 3 and 1 so they become -3 and -1

Inverse of a 2 x 2 matrix

Step 4. Divide each element of the matrix by ∆

Remark: for verification AA-1 = I

which in this case is 1, so no apparent change will be noticed. The inverse of the matrix is then

We use a more general procedure to find the inverse of a 3 x 3 matrix.

1. Augment this matrix with the 3 x 3 identity matrix. 2. Use elementary row operations to transform the matrix

on the left side of the vertical line to the 3 x 3 identity matrix. The row operation is used for the entire row so that the matrix on the right hand side of the vertical line will also change.

3. When the matrix on the left is transformed to the 3 x 3 identity matrix, the matrix on the right of the vertical line is the inverse.

Inverse of a 3 x 3 matrixProcedure

Procedure Inverse of a 3 x 3 matrixProcedure

Here are the necessary row operations: Step 1: Get zeros below the 1 in the first column by

multiplying row 1 by -2 and adding the result to R2. Row 2 is replaced by this sum.

Step2. Multiply R1 by 2, add result to R3 and replace R3 by that result.

Step 3. Multiply row 2 by (1/3) to get a 1 in the second row first position.

Step 4. Add R1 to R2 and replace R1 by that sum.

Step 5. Multiply R2 by 4, add result to R3 and replace R3 by that sum.

Step 6. Multiply R3 by 3/5 to get a 1 in the third row, third position.

Inverse of a 3 x 3 matrixContinuation of Procedure

Step 7. Eliminate the 5/3 in the first row third position by multiplying row 3 by -5/3 and adding result to Row 1.

Step 8. Eliminate the -4/3 in the second row, third position by multiplying R3 by 4/3 and adding result to R2.

Step 9. You now have the identity matrix on the left, which is our goal.

Inverse of a 3 x 3 matrixFinal result

Normal form of a matrix

Where is the unit matrix of order r. hence ρ(A) = r

Square Matrices P & Q of Orders m & n respectively , such that PAQ is in the normal form

Working rule:-1. write A = I A I2. Reduce the matrix on L.H.S.to normal form by

applying elementary row or column operation.Remark : * if row operation is applied on L.H.S. then this

operation is applied on pre-factor of A on R.H.S* if column operation is applied on L.H.S. then this

operation is applied on post-factor of A on R.H.S The matrices P and Q are not unique

Consistent and Inconsistent Systems of Equations

All the systems of equations that we have seen in this section so far have had unique solutions. These are referred to as Consistent Systems of Equations, meaning that for a given system, there exists one solution set for the different variables in the system or infinitely many sets of solution. In other words, as long as we can find a solution for the system of equations, we refer to that system as being consistent

Inconsistent systems arise when the lines or planes formed from the systems of equations don't meet at any point.

Consistency Chart

Eigen values and eigenvectors have their origins in physics, in particular in problems where motion is involved, although their uses extend from solutions to stress and strain problems to differential equations and quantum mechanics. we can use matrices to deform a body - the concept of STRAIN. Eigenvectors are vectors that point in directions where there is no rotation. Eigen values are the change in length of the eigenvector from the original length.

Eigen values and Eigen vectors

Origin of Eigen values and Eigen vectors

Eigen values and Eigen vectors

Let A be an nxn matrix and consider the vector equation:

Ax = x A value of for which this equation has a

solution x≠0 is called an Eigen value of the matrix A.

The corresponding solutions x are called the Eigen vectors of the matrix A.

Solving for Eigen ValuesAx=x

Ax - x = 0(A- I)x = 0

This is a homogeneous linear system, homogeneous meaning that the RHS are all zeros.

For such a system, a theorem states that a solution exists given that det(A- I)=0.

The Eigen values are found by solving the above equation.

Solving for Eigen values cont’

Simple example: find the Eigen values for the matrix:

Eigen values are given by the equation det(A-I) = 0:

So, the roots of the last equation are -1 and -6. These are the Eigen values of matrix A.

22

25A

674)2)(5(

22

25)det(

2

IA

Eigenvectors For each Eigen value, , there is a

corresponding eigenvector, x. This vector can be found by substituting

one of the Eigen values back into the original equation: Ax = x : for the example: -5x1 + 2x2 = x1

2x1 – 2x2 = x2

Using =-1, we get x2 = 2x1, and by arbitrarily choosing x1 = 1, the Eigenvector corresponding to =-1 is:

and similarly,

2

11x

1

22x

Special matrices

A matrix is called symmetric if:AT = A

A skew-symmetric matrix is one for which:

AT = -A An orthogonal matrix is one whose

transpose is also its inverse: AT = A-1