Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
Please go through the slides. It is very interesting way to learn this chapter for 2020-21.If you like this PPT please put a thanks message in my number 9826371828.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
Please go through the slides. It is very interesting way to learn this chapter for 2020-21.If you like this PPT please put a thanks message in my number 9826371828.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Gen AI Study Jams _ For the GDSC Leads in India.pdf
Matrices ppt
1.
2. 11 12 13 14
21 22 23 24
31 32 33 34
1 2 3 4m m m m
a a a a
a a a a
a a a a
a a a a
Row 1
Row 2
Row 3
Row m
Column 1 Column 2 Column 3 Column 4
3. A matrix of m rows and n columns is called
a matrix with dimensions m x n.
2 3 4
1.) 1
1
2
3 8 9
2.) 2 5
6 7 8
10
3.)
7
4.) 3 4
2 X 3
3 X 3
2 X 1
1 X 2
6. To add matrices, we add the corresponding
elements. They must have the same
dimensions.
5 0 6 3
4 1 2 3
A B
A + B
5 6 0 3
4 2 1 3
1 3
6 4
7. 2 1 3 0 0 0
2.)
1 0 1 0 0 0
2 1 3
1 0 1
When a zero matrix is added to another
matrix of the same dimension, that same
matrix is obtained.
12. Scalar Multiplication:
1 2 3
1 2 3
4 5 6
k
We multiply each element of the matrix
by scalar k.
1 2 3
1 2 3
4 5 6
k k k
k k k
k k k
14. • Associative Property of Addition
(A+B)+C = A+(B+C)
• Commutative Property of Addition
A+B = B+A
• Distributive Property of Addition and
Subtraction S(A+B) = SA+SB
S(A-B) = SA-SB
• NOTE: Multiplication is not included!!!
15. • 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:
EROminimum # of rows with at least one nonzero entry
or
ECOminimum # of columns with at least one nonzero entry
16. Math for CS Lecture 2 16
nnnnnn
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
2211
22222121
11212111
nnnnnn
n
n
b
b
b
x
x
x
aaa
aaa
aaa
2
1
2
1
21
22221
11211
(1)
17. bxA
Each side of the equation
bAxAA 11
Can be multiplied by A-1 :
Due to the definition of A-1: xxIxAA 1
Therefore the solution of (2) is:
(2)
bAx 1
18. • A-1 does not exist for every A.
• The linear system of equations A·x=b has a
solution, or said to be consistent if
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).
19. Math for CS Lecture 2 19
5
4
42
21
2
1
x
x
00
21
Rank{A}=1
Rank{A|b}=2 > Rank{A}
ERO:Multiply the first row with
-2 and add to the second row
3
4
0
2
0
1
20. Math for CS Lecture 2 20
• The system has a unique solution if
Rank{A}=Rank{A|b}= n,
where n is the order of the system.
21. Math for CS Lecture 2 21
• If Rank{A}=n
Det{A} 0 A-1 exists Unique solution
2
4
11
21
2
1
x
x
22. • 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
8
4
42
21
2
1
x
x
Consistent so solvable
Rank{A}=Rank{A|b}=1
23. • 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 eigen value (or characteristic value) of
A.
Note: The zero vector can not be an
eigenvector even though A0 = λ0. But λ = 0
can be an eigen value.
Eigenvalues and Eigenvectors
24. 2 2 4
1 3 6
2 4 2 0
:
3 6 1 0
2 0
0, 0
1 0
, , 0 .
Show x is aneigenvector for A
Solution Ax
But for x
Thus xis aneigenvector of A and is aneigenvalue
Example:
25. 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 = 0 or
(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.
26. 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.
51
122
A
)2)(1(23
12)5)(2(
51
122
2
AI
Eigenvalues: examples
200
020
012
A
0)2(
200
020
012
3
AI
27. Example 1 (cont.):
00
41
41
123
)1(:1 AI
0,
1
4
,404
2
1
1
2121
tt
x
x
txtxxx
x
00
31
31
124
)2(:2 AI
0,
1
3
2
1
2
ss
x
x
x
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
28. 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.
0
0
0
000
000
010
)2(
3
2
1
x
x
x
AI x
txsx 31 ,
,
1
0
0
0
0
1
0
3
2
1
ts
t
s
x
x
x
x
Eigenvectors
200
020
012
A
29. 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.
30. 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.
31. 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
,
1
0
0
0
0
1
0
3
2
1
ts
t
s
x
x
x
x
32. LINEAR INDEPENDENCE
• Definition: A set of vectors {v1, …, vp} in
is said to be linearly independent if the vector
equation
has only the trivial solution. The set {v1, …, vp} is
said to be linearly dependent if there exist
weights c1, …, cp, not all zero, such that
n
1 1 2 2
v v ... v 0p p
x x x
1 1 2 2
v v ... v 0p p
c c c
33. • Equation (1) is called a linear dependence
relation among v1, …, vp when the weights are not
all zero. A set is linearly dependent if and only if it
is not linearly independent.
Example 1: Let , , and .
1
1
v 2
3
2
4
v 5
6
3
2
v 1
0
34. a. Determine if the set {v1, v2, v3} is linearly
independent.
b. If possible, find a linear dependence relation
among v1, v2, and v3.
Solution: We must determine if there is a nontrivial
solution of the following equation.
1 2 3
1 4 2 0
2 5 1 0
3 6 0 0
x x x
35. Row operations on the associated augmented matrix
show that
.
x1 and x2 are basic variables, and x3 is free.
Each nonzero value of x3 determines a nontrivial
solution of (1).
Hence, v1, v2, v3 are linearly dependent.
1 4 2 0 1 4 2 0
2 5 1 0 0 3 3 0
3 6 0 0 0 0 0 0
:
36. b. To find a linear dependence relation among v1,
v2, and v3, row reduce the augmented matrix
and write the new system:
• Thus, , , and x3 is free.
• Choose any nonzero value for x3—say, .
• Then and .
1 0 2 0
0 1 1 0
0 0 0 0
1 3
2 3
2 0
0
0 0
x x
x x
1 3
2x x 2 3
x x
3
5x
1
10x 2
5x
37. CAYLEY HAMILTON THEOREM
Every square matrix satisfies its own
characteristic equation.
Let A = [aij]n×n be a square matrix
then,
nnnn2n1n
n22221
n11211
a...aa
................
a...aa
a...aa
A
38. Let the characteristic polynomial of A be (λ)
Then,
The characteristic equation is
11 12 1n
21 22 2n
n1 n2 nn
φ(λ) = A - λI
a - λ a ... a
a a - λ ... a
=
... ... ... ...
a a ... a - λ
| A - λI|=0
39. Note 1:- Premultiplying equation (1) by A-1 , we
have
n n-1 n-2
0 1 2 n
n n-1 n-2
0 1 2 n
We are to prove that
p λ +p λ +p λ +...+p = 0
p A +p A +p A +...+p I= 0 ...(1)
I
n-1 n-2 n-3 -1
0 1 2 n-1 n
-1 n-1 n-2 n-3
0 1 2 n-1
n
0 =p A +p A +p A +...+p +p A
1
A =- [p A +p A +p A +...+p I]
p
40. This result gives the inverse of A in terms of
(n-1) powers of A and is considered as a practical
method for the computation of the inverse of the
large matrices.
Note 2:- If m is a positive integer such that m > n
then any positive integral power Am of A is linearly
expressible in terms of those of lower degree.
41. Verify Cayley – Hamilton theorem for the matrix
A = . Hence compute A-1 .
Solution:- The characteristic equation of A is
211
121
112
tion)simplifica(on049λ6λλor
0
λ211
1λ21
11λ2
i.e.,0λIA
23
Example 1:-
45. 45
Given find Adj A by using Cayley –
Hamilton theorem.
Solution:- The characteristic equation of the given
matrix A is
113
110
121
A
tion)simplifica(on035λ3λλor
0
λ113
1λ10
1-2λ1
i.e.,0λIA
23
Example 2:-
49. 49
DIAGONALISATION OF A
MATRIX
Diagonalisation of a matrix A is the process of
reduction A to a diagonal form.
If A is related to D by a similarity transformation,
such that D = M-1AM then A is reduced to the
diagonal matrix D through modal matrix M. D is
also called spectral matrix of A.
50. 50
REDUCTION OF A MATRIX TO
DIAGONAL FORM
If a square matrix A of order n has n linearly
independent eigen vectors then a matrix B can
be found such that B-1AB is a diagonal matrix.
Note:- The matrix B which diagonalises A is called
the modal matrix of A and is obtained by
grouping the eigen vectors of A into a square
matrix.
51. 51
Similarity of matrices:-
A square matrix B of order n is said to be a
similar to a square matrix A of order n if
B = M-1AM for some non singular
matrix M.
This transformation of a matrix A by a non –
singular matrix M to B is called a similarity
transformation.
Note:- If the matrix B is similar to matrix A, then B
has the same eigen values as A.
52. 52
Reduce the matrix A = to diagonal form by
similarity transformation. Hence find A3.
Solution:- Characteristic equation is
=> λ = 1, 2, 3
Hence eigen values of A are 1, 2, 3.
300
120
211
0
λ-300
1λ-20
21λ1-
Example:-
53. 53
Corresponding to λ = 1, let X1 = be the eigen
vector then
3
2
1
x
x
x
0
0
1
kX
x0x,kx
02x
0xx
02xx
0
0
0
x
x
x
200
110
210
0X)I(A
11
3211
3
32
32
3
2
1
1
54. 54
Corresponding to λ = 2, let X2 = be the eigen
vector then,
3
2
1
x
x
x
0
1-
1
kX
x-kx,kx
0x
0x
02xxx
0
0
0
x
x
x
100
100
211-
0X)(A
22
32221
3
3
321
3
2
1
2
0,
I2
55. 55
Corresponding to λ = 3, let X3 = be the eigen
vector then,
3
2
1
x
x
x
2
2-
3
kX
xk-x,kx
0x
02xxx
0
0
0
x
x
x
000
11-0
212-
0X)(A
33
13332
3
321
3
2
1
3
3
2
2
3
,
2
I3
k
x
59. 59
ORTHOGONAL TRANSFORMATION
OF A SYMMETRIC MATRIX TO
DIAGONAL FORM
A square matrix A with real elements is said to
be orthogonal if AA’ = I = A’A.
But AA-1 = I = A-1A, it follows that A is orthogonal if
A’ = A-1.
Diagonalisation by orthogonal transformation is
possible only for a real symmetric matrix.
60. 60
If A is a real symmetric matrix then eigen
vectors of A will be not only linearly independent
but also pairwise orthogonal.
If we normalise each eigen vector and use
them to form the normalised modal matrix N then it
can be proved that N is an orthogonal matrix.
61. 61
The similarity transformation M-1AM = D takes
the form N’AN = D since N-1 = N’ by a property of
orthogonal matrix.
Transforming A into D by means of the
transformation N’AN = D is called as orthogonal
reduction or othogonal transformation.
Note:- To normalise eigen vector Xr, divide each
element of Xr, by the square root of the sum of the
squares of all the elements of Xr.
62. 62
Diagonalise the matrix A = by means of an
orthogonal transformation.
Solution:-
Characteristic equation of A is
204
060
402
66,2,λ
0λ)16(6λ)λ)(2λ)(6(2
0
λ204
0λ60
40λ2
Example :-
63. 63
I
1
1 2
3
1
1
2
3
1 3
2
1 3
1 1 2 3 1
1 1
x
whenλ = -2,let X = x betheeigenvector
x
then (A + 2 )X = 0
4 0 4 x 0
0 8 0 x = 0
4 0 4 x 0
4x + 4x = 0 ...(1)
8x = 0 ...(2)
4x + 4x = 0 ...(3)
x = k ,x = 0,x = -k
1
X = k 0
-1
64. 64
2
2I
0
1
2
3
1
2
3
1 3
1 3
1 3 2
2 2 3
x
whenλ = 6,let X = x betheeigenvector
x
then (A -6 )X = 0
-4 0 4 x 0
0 0 x = 0
4 0 -4 x 0
4x +4x = 0
4x - 4x = 0
x = x and x isarbitrary
x must be so chosen that X and X are orthogonal among th
.1
emselves
and also each is orthogonal with X
65. 65
2 3
3 1
3 2
3
1 α
Let X = 0 and let X = β
1 γ
Since X is orthogonal to X
α - γ = 0 ...(4)
X is orthogonal to X
α + γ = 0 ...(5)
Solving (4)and(5), we get α = γ = 0 and β is arbitrary.
0
Taking β =1, X = 1
0
1 1 0
Modal matrix is M = 0 0 1
-1 1
0