2. MOHAMMAD IMRAN DEPARTMENT OF APPLIED SCIENCES JAHANGIRABAD EDUCATIONAL GROUP OF INSTITUTES

3. 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

4. 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 solution Inverse of a matrix Normal form of the matrix Rank of matrix by using the normal form Non-singular matrices P & Q which makes normal form with given matrix A as PAQ

5. Outline cont’  Consistency  Eigen values and Eigenvectors

6. 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:

7. 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.

8. 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.

9. Matrix Addition Multiplication of a Matrix by a Scalar Matrix Multiplication Matrix Transposition Finding the Determinate of a Matrix Matrix Inversion Matrix Operations

10. Matrix must be the same size in order to add Matrix addition is commutative: A + B = B + A Matrix Addition

11. 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:

12. 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

13. 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

14. 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

15. Example – Matrix Multiplication  Find  We know that matrix C will be 2 × 2  Element c 11 is found by multiplying terms of row 1 of A and column 1 of B:

16. Example – Matrix Multiplication Element c 12 is found by multiplying terms of row 1 of A and column 2 of B:

17. Example – Matrix Multiplication Element c 21 is found by multiplying terms of row 2 of A and column 1 of B:

18. Example – Matrix Multiplication Element c 22 is found by multiplying terms of row 2 of A and column 2 of B:

19. Example – Matrix Multiplication Solution:

20. Matrix Multiplication In general, matrix multiplication is not commutative: AB ≠ BA

21. 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:

22. 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....n3567....n

23.  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

24.  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

25. 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

26.  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

27.  Null Matrix : A matrix whose all elements are zero called “Null Matrix”. Remark: This matrix is also type of square matrix. Types of Matrices

28.  Symmetric Matrix : A matrix which is equal to its transpose said to be “Symmetric Matrix” A = We can see that A =A T Types of Matrices

29.  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 = - A T Types of Matrices

30.  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

31.  Upper Triangular matrix :- if all the elements above the diagonal are zero then this type of matrix is called “Upper triangular matrix” For Ex.

32.  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

33.  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

34.  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

35. Orthogonal matrix :- An orthogonal matrix is one whose transpose is also its inverse. A T = A -1 Types of Matrices

36. 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:

37. Determinate of a Matrix 3 × 3:

38. 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.

39. Matrix inverse  The inverse of the matrix A is denoted as A -1  By definition, AA -1 = A -1 A = 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.

40. 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

41.  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

42. 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

43. 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

44. 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 matrix Procedure

45. Procedure Inverse of a 3 x 3 matrix Procedure 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 R 2. Row 2 is replaced by this sum.  Step2. Multiply R 1 by 2, add result to R 3 and replace R 3 by that result.  Step 3. Multiply row 2 by (1/3) to get a 1 in the second row first position.

46.  Step 4. Add R 1 to R 2 and replace R 1 by that sum.  Step 5. Multiply R 2 by 4, add result to R 3 and replace R 3 by that sum.  Step 6. Multiply R 3 by 3/5 to get a 1 in the third row, third position. Inverse of a 3 x 3 matrix Continuation of Procedure

47.  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 R 3 by 4/3 and adding result to R 2.  Step 9. You now have the identity matrix on the left, which is our goal. Inverse of a 3 x 3 matrix Final result

48. Normal form of a matrix Where is the unit matrix of order r. hence ρ(A) = r

49. Square Matrices P & Q of Orders m & n respectively, such that PAQ is in the normal form Working rule:- 1. write A = I A I 2. 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

50. 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.

51. Consistency Chart

52.  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

53. 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.

54. Solving for Eigen Values Ax= 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.

55. 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.

56. 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:-5x 1 + 2x 2 = x 1 2x 1 – 2x 2 = x 2  Using =-1, we get x 2 = 2x 1, and by arbitrarily choosing x 1 = 1, the Eigenvector corresponding to =-1 is: and similarly,

57. Special matrices  A matrix is called symmetric if: A T = A  A skew-symmetric matrix is one for which: A T = -A  An orthogonal matrix is one whose transpose is also its inverse: A T = A -1