This document discusses eigen values and eigenvectors. It defines eigen values and eigenvectors as scalars (eigenvalues) and vectors (eigenvectors) that satisfy the equation Ax = λx, where A is a matrix and λ is an eigenvalue. It provides properties of eigenvalues, including that the sum of eigenvalues equals the trace of A. It also discusses algebraic and geometric multiplicity, the characteristic equation, Cayley-Hamilton theorem, and examples to illustrate these concepts.
this the slide for learning derogatory and non derogatory matrices. You can eaily refer this. we have also stated some basic learnt topics in matrices so that you won't confusing over the steps of example. this is the part of linear algebra. the slide template is taken from slidesgo.com but it is designed by me. we have also mentioned the polynomials that we have to find and the necessary steps that we need to follow to show or prove whether the matrix is derogatory or not. i hope you like this slide.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
this the slide for learning derogatory and non derogatory matrices. You can eaily refer this. we have also stated some basic learnt topics in matrices so that you won't confusing over the steps of example. this is the part of linear algebra. the slide template is taken from slidesgo.com but it is designed by me. we have also mentioned the polynomials that we have to find and the necessary steps that we need to follow to show or prove whether the matrix is derogatory or not. i hope you like this slide.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
TOP 10 B TECH COLLEGES IN JAIPUR 2024.pptxnikitacareer3
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NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
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3. Eigenvalues and Eigenvectors
• If A is an n x n matrix and λ is a scalar for which Ax = λx has a nontrivial
solution x ∈ ℜⁿ, then λ is an eigenvalue of A and x is a corresponding
eigenvector of A.
– Ax=λx=λIx
– (A-λI)x=0
• The matrix (A-λI ) is called the characteristic matrix of a where I is the
Unit matrix.
–
• The equation det (A-λI )= 0 is called characteristic equation of A and the
roots of this equation are called the eigenvalues of the matrix A.The set
of all eigenvectors is called the eigenspace of A corresponding to λ.The
set of all eigenvalues of a is called spectrum of A.
4. Characteristic Equation
▪ If A is any square matrix of order n, we can form the matrix , where
is the nth order unit matrix.
▪ The determinant of this matrix equated to zero,
▪ i.e.,
is called the characteristic equation of A.
0
λ
a
...
a
a
...
...
...
...
a
...
λ
a
a
a
...
a
λ
a
λ
A
nn
n2
n1
2n
22
21
1n
12
11
I
5. • On expanding the determinant, we get
• where k’s are expressible in terms of the elements a
•The roots of this equation are called Characteristic roots
or latent roots or eigen values of the matrix A.
•X = is called an eigen vector or latent vector
0
k
...
λ
k
λ
k
λ
1)
( n
2
n
2
1
n
1
n
n
4
2
1
...
x
x
x
6. Properties of Eigen Values:-
1. The sum of the eigen values of a matrix is the sum of the
elements of the principal diagonal.
2.The product of the eigen values of a matrix A is equal to its
determinant.
3. If is an eigen value of a matrix A, then 1/ is the eigen value
of A-1 .
4.If is an eigen value of an orthogonal matrix, then 1/ is also its
eigen value.
7. PROPERTY 1:- If λ1, λ2,…, λn are the eigen values of A, then
i. k λ1, k λ2,…,k λn are the eigen values of the matrix kA,
where k is a non – zero scalar.
ii. are the eigen values of the inverse
matrix A-1.
iii. are the eigen values of Ap, where p is any positive
integer.
8. Algebraic & Geometric Multiplicity
▪ If the eigenvalue λ of the equation det(A-λI)=0 is repeated n times
then n is called the algebraic multiplicity of λ.The number of linearly
independent eigenvectors is the difference between the number of
unknowns and the rank of the corresponding matrix A- λI and is
known as geometric multiplicity of eigenvalue λ.
9. Cayley-Hamilton Theorem:-
• Every square matrix satisfies its own characteristic equation.
• Let A = [aij]n×n be a square matrix then,
n
n
nn
2
n
1
n
n
2
22
21
n
1
12
11
a
...
a
a
....
....
....
....
a
...
a
a
a
...
a
a
A
10. 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
11. 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)
Note 1:- Premultiplying equation (1) by A-1 , we
have
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
12. 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.
13. Example 1:-
Verify Cayley – Hamilton theorem for the matrix
A = . Hence compute A-1 .
Solution:-The characteristic equation of A is
2
1
1
1
2
1
1
1
2
tion)
simplifica
(on
0
4
9λ
6λ
λ
or
0
λ
2
1
1
1
λ
2
1
1
1
λ
2
i.e.,
0
λI
A
2
3
21. Similarity of Matrix
▪ IfA & B are two square matrices of order n then B is said to be similar
to A, if there exists a non-singular matrix P such that,
B= P-1AP
1. Similarity matrices is an equivalence relation.
2. Similarity matrices have the same determinant.
3. Similar matrices have the same characteristic polynomial and
hence the same eigenvalues. If x is an eigenvector corresponding to
the eigenvalue λ, then P-1x is an eigenvector of B corresponding to
the eigenvalue λ where B= P-1AP.
22. Diagonalization
▪ A matrix A is said to be diagonalizable if it is similar to diagonal
matrix.
▪ A matrix A is diagonalizable if there exists an invertible matrix P such
that P-1AP=D where D is a diagonal matrix, also known as spectral
matrix.The matrix P is then said to diagonalize A of transform A to
diagonal form and is known as modal matrix.
23. 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 ofA into a square
matrix.
24. Example:-
Reduce the matrix A = to diagonal form by
similarity transformation. Hence find A3.
Solution:- Characteristic equation is
=> λ = 1, 2, 3
Hence eigenvalues of A are 1, 2, 3.
3
0
0
1
2
0
2
1
1
0
λ
-
3
0
0
1
λ
-
2
0
2
1
λ
1-
25. Corresponding to λ = 1, let X1 = be the eigen
vector then
3
2
1
x
x
x
0
0
1
k
X
x
0
x
,
k
x
0
2x
0
x
x
0
2x
x
0
0
0
x
x
x
2
0
0
1
1
0
2
1
0
0
X
)
I
(A
1
1
3
2
1
1
3
3
2
3
2
3
2
1
1
26. Corresponding to λ = 2, let X2 = be the eigen
vector then,
3
2
1
x
x
x
0
1
-
1
k
X
x
-k
x
,
k
x
0
x
0
x
0
2x
x
x
0
0
0
x
x
x
1
0
0
1
0
0
2
1
1
-
0
X
)
(A
2
2
3
2
2
2
1
3
3
3
2
1
3
2
1
2
0
,
I
2
27. Corresponding to λ = 3, let X3 = be the eigen
vector then,
3
2
1
x
x
x
2
2
-
3
k
X
x
k
-
x
,
k
x
0
x
0
2x
x
x
0
0
0
x
x
x
0
0
0
1
1
-
0
2
1
2
-
0
X
)
(A
3
3
1
3
3
3
2
3
3
2
1
3
2
1
3
3
2
2
3
,
2
I
3
k
x
31. Orthogonally Similar Matrices
▪ If A & B are two square matrices of order n then B is said to be orthogonally
similar to A, if there exists orthogonal matrix P such that
B= P-1AP
Since P is orthogonal,
P-1=PT
B= P-1AP=PTAP
1. A real symmetric of order n has n mutually orthogonal real eigenvectors.
2. Any two eigenvectors corresponding to two distinct eigenvalues of a real
symmetric matrix are orthogonal.
32. Diagonalises the matrix A = by means of an
orthogonal transformation.
Solution:-
Characteristic equation of A is
32
Example :-
2
0
4
0
6
0
4
0
2
6
6,
2,
λ
0
λ)
16(6
λ)
λ)(2
λ)(6
(2
0
λ
2
0
4
0
λ
6
0
4
0
λ
2
33. I
1
1 2
3
1
1
2
3
1 3
2
1 3
1 1 2 3 1
1 1
x
when λ = -2,let X = x be the eigen vector
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
34. 2
2
I
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
35.
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
37. Quadratic Forms
DEFINITION:-
A homogeneous polynomial of second degree in any number of
variables is called a quadratic form.
For example,
ax2 + 2hxy +by2
ax2 + by2 + cz2 + 2hxy + 2gyz + 2fzx and
ax2 + by2 + cz2 + dw2 +2hxy +2gyz + 2fzx + 2lxw + 2myw + 2nzw
are quadratic forms in two, three and four variables
38. In n – variables x1,x2,…,xn, the general quadratic form
is
In the expansion, the co-efficient of xixj = (bij + bji).
38
n
1
j
n
1
i
ji
ij
j
i
ij b
b
where
,
x
x
b
).
b
(b
2
1
a
where
x
x
a
x
x
b
b
a
and
a
a
where
b
b
2a
Suppose
ji
ij
ij
j
i
n
1
j
n
1
i
ij
j
i
n
1
j
n
1
i
ij
ii
ii
ji
ij
ij
ij
ij
39. Hence every quadratic form can be written as
get
we
form,
matrix
in
forms
quadratic
of
examples
said
above
the
writing
Now
.
x
,...,
x
,
x
X
and
a
A
where
symmetric,
always
is
A
matrix
the
that
so
AX,
X'
x
x
a
n
2
1
ij
j
i
n
1
j
n
1
i
ij
y
x
b
h
h
a
y]
[x
by
2hxy
ax
(i) 2
2
41. Two Theorems On Quadratic Form
Theorem(1): A quadratic form can always be expressed with respect to
a given coordinate system as
where A is a unique symmetric matrix.
Theorem2: Two symmetric matrices A and B represent the same
quadratic form if and only if
B=PTAP
where P is a non-singular matrix.
Ax
x
Y T
42. Nature of Quadratic Form
A real quadratic form X’AX in n variables is said to be
i. Positive definite if all the eigen values ofA > 0.
ii. Negative definite if all the eigen values of A < 0.
iii. Positive semi definite if all the eigen values ofA 0 and at least one
eigen value = 0.
iv. Negative semi definite if all the eigen values of
A 0 and at least one eigen value = 0.
v. Indefinite if some of the eigen values ofA are + ve and others – ve.
43. Find the nature of the following quadratic forms
i. x2 + 5y2 + z2 + 2xy + 2yz + 6zx
ii. 3x2 + 5y2 + 3z2 – 2yz + 2zx – 2xy
Solution:-
i. The matrix of the quadratic form is
43
Example :-
1
1
3
1
5
1
3
1
1
A
44. The eigen values of A are -2, 3, 6.
Two of these eigen values being positive and one being
negative, the given quadratric form is indefinite.
ii. The matrix of the quadratic form is
The eigen values of A are 2, 3, 6. All these eigen values being
positive, the given quadratic form is positive definite.
3
1
1
1
5
1
1
1
3
A
45. Linear Transformation of a Quadratic
Form
▪ Let X’AX be a quadratic form in n- variables and let X = PY ….. (1)
where P is a non – singular matrix, be the non – singular
transformation.
▪ From (1), X’ = (PY)’ =Y’P’ and hence
X’AX =Y’P’APY =Y’(P’AP)Y
=Y’BY …. (2)
where B = P’AP.
46. Therefore,Y’BY is also a quadratic form in n- variables. Hence it
is a linear transformation of the quadratic form X’AX under the
linear transformation X = PY and B = P’AP.
Note. (i) Here B = (P’AP)’ = P’AP = B
(ii) ρ(B) = ρ(A)
Therefore, A and B are congruent matrices.
47. Reduce 3x2 + 3z2 + 4xy + 8xz + 8yz into canonical form.
Or
Diagonalises the quadratic form 3x2 + 3z2 + 4xy + 8xz + 8yz by
linear transformations and write the linear transformation.
Or
Reduce the quadratic form 3x2 + 3z2 + 4xy + 8xz + 8yz into the
sum of squares.
47
Example:-
48. Solution:- The given quadratic form can be written as X’AX where
X = [x, y, z]’ and the symmetric matrix
A =
Let us reduce A into diagonal matrix. We know tat A = I3AI3.
3
4
4
4
0
2
4
2
3
1
0
0
0
1
0
0
0
1
3
4
4
4
0
2
4
2
3
1
0
0
0
1
0
0
0
1
3
4
4
4
0
2
4
2
3
51. The canonical form of the given quadratic form is
Here ρ(A) = 3, index = 1, signature = 1 – (2) = -1.
Note:- In this problem the non-singular transformation which
reduces the given quadratic form into the canonical form is X = PY.
i.e.,
2
3
2
2
2
1
3
2
1
3
2
1
y
y
3
4
3y
y
y
y
1
0
0
0
3
4
0
0
0
3
y
y
y
AP)Y
(P'
Y'
3
2
1
1
1
2
0
1
3
2
0
0
1
y
y
y
z
y
x