Transcript of MOHAMMAD IMRAN DEPARTMENT OF APPLIED SCIENCES JAHANGIRABAD EDUCATIONAL GROUP OF INSTITUTES.
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- MOHAMMAD IMRAN DEPARTMENT OF APPLIED SCIENCES JAHANGIRABAD
EDUCATIONAL GROUP OF INSTITUTES
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- 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
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- 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
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- Outline cont Consistency Eigen values and Eigenvectors
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- 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:
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- The dimension (size) of a matrix is defined by the number of
rows and number of columns Examples: 3 3: 24: Review: Properties of
Matrices cont.
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- 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.
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- Matrix Addition Multiplication of a Matrix by a Scalar Matrix
Multiplication Matrix Transposition Finding the Determinate of a
Matrix Matrix Inversion Matrix Operations
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- Matrix must be the same size in order to add Matrix addition is
commutative: A + B = B + A Matrix Addition
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- 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:
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- 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
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- 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
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- 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
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- 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:
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- Example Matrix Multiplication Element c 12 is found by
multiplying terms of row 1 of A and column 2 of B:
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- Example Matrix Multiplication Element c 21 is found by
multiplying terms of row 2 of A and column 1 of B:
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- Example Matrix Multiplication Element c 22 is found by
multiplying terms of row 2 of A and column 2 of B:
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- Example Matrix Multiplication Solution:
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- Matrix Multiplication In general, matrix multiplication is not
commutative: AB BA
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- 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:
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- 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
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- 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
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- 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
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- 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
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- 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
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- Null Matrix : A matrix whose all elements are zero called Null
Matrix. Remark: This matrix is also type of square matrix. Types of
Matrices
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- 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
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- 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
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- 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
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- Upper Triangular matrix :- if all the elements above the
diagonal are zero then this type of matrix is called Upper
triangular matrix For Ex.
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- 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
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- 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
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- 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
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- Orthogonal matrix :- An orthogonal matrix is one whose
transpose is also its inverse. A T = A -1 Types of Matrices
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- 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:
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- Determinate of a Matrix 3 3:
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- 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.
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- 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.
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- 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
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- 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
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- Inverse of a 2 x 2 matrix Step 2. Reverse the entries of the
main diagonal consisting of the two 2s. 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
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- 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
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- 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
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- 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.
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- 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
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- 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
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- Normal form of a matrix Where is the unit matrix of order r.
hence (A) = r
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- 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
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- 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.
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- Consistency Chart
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- 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
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- 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 x0 is called an Eigen value of the matrix
A. The corresponding solutions x are called the Eigen vectors of
the matrix A.
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- 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.
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- 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.
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- 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,
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- 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