This document provides an overview of matrices including:
- How to describe matrices using m rows and n columns
- Common types of matrices such as row, column, zero, square, diagonal, and unit matrices
- Basic matrix operations including addition, subtraction, scalar multiplication
- Rules for matrix multiplication including that matrices must be conformable
- The transpose of a matrix which is obtained by interchanging rows and columns
- Properties of transposed matrices including (A+B)T = AT + BT and (AB)T = BTAT
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.
Matrix theory" redirects here. For the physics topic, see Matrix string theory.
An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of the matrix.
In mathematics, a matrix (plural matrices) is a rectangular array[1] (see irregular matrix) of numbers, symbols, or expressions, arranged in rows and columns.[2][3] For example, the dimension of the matrix below is 2 × 3 (read "two by three"), because there are two rows and three columns:
{\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}.}{\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}.}
Provided that they have the same size (each matrix has the same number of rows and the same number of columns as the other), two matrices can be added or subtracted element by element (see conformable matrix). The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second (i.e., the inner dimensions are the same, n for an (m×n)-matrix times an (n×p)-matrix, resulting in an (m×p)-matrix). There is no product the other way round, a first hint that matrix multiplication is not commutative. Any matrix can be multiplied element-wise by a scalar from its associated field.
Matrix Arithmetic and Properties
Some Special Matrices
Nonsingular Matrices
Symmetric Matrices
Orthogonal and Orthonormal Matrix
Complex Matrices
Hermitian Matrices
MATRICES maths project.pptxsgdhdghdgf gr to f HR fpremkumar24914
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The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
2. Definition
How to describe matrices ?
Types of matrices
Operation in matrices
Transpose of matrices
References
3. Matrices is a set or group of numbers
arrange in a square or rectangular
array enclose by two brackets.
Matrices can be written as an m*n
matrix.
‘m’ is horizontal lines called rows.
‘n’ is vertical line called columns.
4. An element occurring in the ith row and
jth column of a matrix A will be called
the (i , j)th element of A to be denoted
by aij
mnijmm
nij
inij
aaaa
aaaa
aaaa
21
22221
1211
...
...
Amxn=
7. Row matrix
Column matrix
Zero or null matrix
Square matrix
Diagonal matrix
Scalar matrix
Unit matrix
Comparable matrix
Equal matrix
8. A matrix having only one row is known as
a row matrix or a row vector.
Example:
• A= [5 18] is a row matrix of order 1x2.
• B= [ 2 4 3 7] is a row matrix of order
1x4.
9. A matrix having only one column is
known as column matrix or a column
vector.
Example:
2
4
1
3
1
1
21
11
ma
a
a
10. A matrix each of whose elements is
zero is called a zero is called zero or
null matrix.
Example:
0
0
0
000
000
000
0ija For all i,j
11. A matrix having the same number of
rows and column is called a square
matrix.
Example:
03
11
166
099
111
12. A square matrix in which every non-
diagonal element is zero is called a
diagonal matrix.
Example:
100
020
001
9000
0500
0030
0003
i.e. aij =0 for all i = j
aij = 0 for some or all i = j
13. A square matrix in which every non-
diagonal elements is 0 and every
diagonal elements is 1 is called a unit
matrix or an identity matrix.
Example:
1000
0100
0010
0001
10
01
i.e. aij =0 for all i = j
aij = 1 for some or all i = j
ij
ij
a
a
0
0
14.
15. ADDITION AND SUBTRACTION OF MATRICES
The sum or difference of two matrices, A and B of
the same size yields a matrix C of the same size
ijijij bac
Matrices of different sizes cannot be added or
subtracted
16. Commutative Law:
A + B = B + A
Associative Law:
A + (B + C) = (A + B) + C = A + B + C
972
588
324
651
652
137
A
2x3
B
2x3
C
2x3
17. A + 0 = 0 + A = A
A + (-A) = 0 (where –A is the matrix composed of –aij as
elements)
122
225
801
021
723
246
18. SCALAR MULTIPLICATION OF MATRICES
Matrices can be multiplied by a scalar (constant or
single element)
Let k be a scalar quantity; then
kA = Ak
Ex. If k=4 and
14
32
12
13
A
20. MULTIPLICATION OF MATRICES
The product of two matrices is another matrix
Two matrices A and B must be conformable for
multiplication to be possible
i.e. the number of columns of A must equal the number
of rows of B
Example.
A x B = C
(1x3) (3x1) (1x1)
21. B x A = Not possible!
(2x1) (4x2)
A x B = Not possible!
(6x2) (6x3)
Example
A x B = C
(2x3) (3x2) (2x2)
24. Assuming that matrices A, B and C are conformable for the
operations indicated, the following are true:
1. AI = IA = A
2. A(BC) = (AB)C = ABC - (associative law)
3. A(B+C) = AB + AC - (first distributive law)
4. (A+B)C = AC + BC - (second distributive law)
Caution!
1. AB not generally equal to BA, BA may not be
conformable
2. If AB = 0, neither A nor B necessarily = 0
3. If AB = AC, B not necessarily = C
26. Let A be an m*n matrix. Then, the n*m
matrix, obtained by interchanging the
rows and columns of A, to be denoted
by A’. Thus,
If the order of A is m*n then the order
of A’ is n*m
(i,j)th element of A= (j,i)th element of
A’
27. A square matrix A is said to be
symmetric if A’ = A.
Example:
db
ba
A
db
ba
A
T
28. A square matrix A is said to be skew –
symmetric if A’= -A.
Example:
A= A’= = -A
015
503
530
015
103
530
29. TRANSPOSE OF A MATRIX
If :
135
7423
2AA
2x3
17
34
52
3
2
TT
AA
Then transpose of A, denoted AT is:
T
jiij aa For all i and j
30. To transpose:
Interchange rows and columns
The dimensions of AT are the reverse of the dimensions of A
135
7423
2AA
17
34
52
2
3
TT
AA
2 x 3
3 x 2
31. Properties of transposed matrices:
1. (A+B)T = AT + BT
2. (AB)T = BT AT
3. (kA)T = kAT
4. (AT)T = A
