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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