The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
Here is a basic Linear Algebra review for the class of Machine Learning. This is actually becoming a new class in the mathematics of Intelligent Systems, there I will be teaching stuff in
1.- Linear Algebra - From the basics to the Cayley-Hamilton Theorem with applications
2.- Mathematical Analysis - from set to the Reimann Integral
3.- Topology - Mostly in Hilbert Spaces
4.- Optimization - Convex functions, KKT conditions, Duality Theory, etc.
The stuff is going to be interesting...
1. Linear Algebra for Machine Learning: Linear SystemsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the first part which is giving a short overview of matrices and discussing linear systems.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
3. Linear Algebra for Machine Learning: Factorization and Linear TransformationsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the third part which is discussing factorization and linear transformations.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here is a basic Linear Algebra review for the class of Machine Learning. This is actually becoming a new class in the mathematics of Intelligent Systems, there I will be teaching stuff in
1.- Linear Algebra - From the basics to the Cayley-Hamilton Theorem with applications
2.- Mathematical Analysis - from set to the Reimann Integral
3.- Topology - Mostly in Hilbert Spaces
4.- Optimization - Convex functions, KKT conditions, Duality Theory, etc.
The stuff is going to be interesting...
1. Linear Algebra for Machine Learning: Linear SystemsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the first part which is giving a short overview of matrices and discussing linear systems.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
3. Linear Algebra for Machine Learning: Factorization and Linear TransformationsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the third part which is discussing factorization and linear transformations.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
These Slides are very usefull interms of engineering and as well as in other fields of Study .. These are Related with linear Algebra and there Properties Methods to find out the unknowns from the equation...
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
These Slides are very usefull interms of engineering and as well as in other fields of Study .. These are Related with linear Algebra and there Properties Methods to find out the unknowns from the equation...
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
5. Linear Algebra for Machine Learning: Singular Value Decomposition and Prin...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fifth part which is discussing singular value decomposition and principal component analysis.
Here are the slides of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
Here are the slides of the fourth part which is discussing eigenvalues and eigenvectors.
https://www.slideshare.net/CeniBabaogluPhDinMat/4-linear-algebra-for-machine-learning-eigenvalues-eigenvectors-and-diagonalization
Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. 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. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
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A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
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4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
1. Seminar Series on
Linear Algebra for Machine Learning
Part 4: Eigenvalues, Eigenvectors and
Diagonalization
Dr. Ceni Babaoglu
Data Science Laboratory
Ryerson University
cenibabaoglu.com
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
2. Overview
1 Eigenvalues and eigenvectors
2 Some properties of eigenvalues and eigenvectors
3 Similar matrices
4 Diagonalizable matrices
5 Some properties of diagonalizable matrices
6 References
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
3. Eigenvalues and Eigenvectors
When a matrix multiplies a vector in general the magnitude
and direction of the vector will change.
There are special vectors where only their magnitude is
changed when multiplied by a matrix.
These special vectors are called eigenvectors. The value by
which the length changes is the associated eigenvalue.
We say that x is an eigenvector of A if
Ax = λx.
λ is called the associated eigenvalue.
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
4. Eigenvalues and Eigenvectors
The matrix A transforms 5 different vectors to other 5
different vectors. Vector (1) is transformed to vector (a),
vector (2) is transformed to vector (b) and so on.
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
5. Eigenvalues and Eigenvectors
All the vectors except vector (4) change both their magnitude
and direction when transformed by A.
Vector (4) changes only magnitude and does not change
direction.
Vector (4) is an eigenvector of A.
λ =
Magnitude of vector (d)
Magnitude of vector (4)
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
6. Eigenvalues and Eigenvectors
In this mapping the red arrow changes direction but the blue
arrow does not. The blue arrow is an eigenvector of this
mapping because it does not change direction, and since its
length is unchanged, its eigenvalue is 1.
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
7. Some properties
If λ1, λ2, · · · , λn are distinct eigenvalues of a matrix, then the
corresponding eigenvectors e1, e2, · · · , en are linearly
independent.
If e1 is an eigenvector of a matrix with corresponding
eigenvalue λ1, then any nonzero scalar multiple of e1 is also
an eigenvector with eigenvalue λ1.
A real, symmetric square matrix has real eigenvalues, with
orthogonal eigenvectors (can be chosen to be orthonormal).
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
8. Some properties
The equation Ax = λx can be written in the form
(A − λI)x = 0 (1)
λ is an eigenvalue of A if and only if (1) has a nontrivial
solution.
(1) has nontrivial solution if and only if A − λI is singular, or
equivalently
det(A − λI) = 0 → characteristic equation for A (2)
If (2) is expanded, we obtain an nth degree polynomial in λ,
p(λ) = det(A − λI) → characteristic polynomial
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
9. Similar Matrices
A matrix B is said to be similar to a matrix A if there exists a
nonsingular matrix S such that
B = S−1
AS.
For n × n matrices A and B, if A is similar to B, then the two
matrices both have the same characteristic polynomial and,
consequently, both have the same eigenvalues.
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
10. Diagonalizable Matrices
An n × n matrix A is said to be diagonalizable if there exists a
nonsingular matrix X and a diagonal matrix D such that
X−1
AX = D.
We say that X diagonalizes A.
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
11. Some Properties
An n × n matrix A is diagonalizable if and only if A has n
linearly independent eigenvectors.
If A is diagonalizable, then the column vectors of the
diagonalizing matrix X are eigenvectors of A and the diagonal
elements of D are the corresponding eigenvalues of A.
The diagonalizing matrix X is not unique. Reordering the
columns of a given diagonalizing matrix X or multiplying them
by nonzero scalars will produce a new diagonalizing matrix.
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
12. Some Properties
If an n × n matrix A has n distinct eigenvalues, then A is
diagonalizable.
If the eigenvalues are not distinct, then A may or may not be
diagonalizable, depending on whether A has n linearly
independent eigenvectors.
If A is diagonalizable, then A can be factored into a product
XDX−1
.
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
13. Some Properties
All the roots of the characteristic polynomial of a symmetric
matrix are real numbers.
If A is a symmetric matrix, then eigenvectors that belong to
distinct eigenvalues of A are orthogonal.
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
15. Example
A =
2 −2 3
1 1 1
1 3 −1
We solve the equation |A − λI| = 0.
Eigenvalues: λ1 = 1, λ2 = 3, λ3 = −2.
The 3 × 3 matrix A has 3 distinct real eigenvalues.
It has 3 linearly independent eigenvectors which implies A is
diagonalizable.
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
16. Example
To find the eigenvector corresponding to λ1 = 1, we solve the
homogeneous system (A − λ1I) u = 0.
(A − I) u = 0 ⇒
1 −2 3
1 0 1
1 3 −2
·
u1
u2
u3
= 0
⇒
1 −2 3
1 0 1
1 3 −2
R2−R1→R2
R3−R1→R3
−−−−−−−→
1 −2 3
0 2 −2
0 5 −5
R2/2→R2
R3/5→R3
−−−−−→
1 −2 3
0 1 −1
0 1 −1
R1+2R2→R1
R3−R2→R3
−−−−−−−→
1 0 1
0 1 −1
0 0 0
⇒ u1 + u3 = 0 u2 − u3 = 0
⇒ u1 = −u2 = −u3 = α
An eigenvector corresponding to λ1 = 1 is
−1
1
1
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
17. Example
To find the eigenvector corresponding to λ2 = 3, we solve the
homogeneous system (A − λ2I) v = 0.
⇒ (A − 3I) v = 0
⇒
−1 −2 3
1 −2 1
1 3 −4
·
v1
v2
v3
= 0
⇒
−1 −2 3
1 −2 1
1 3 −4
R2+R1→R2
R3+R1→R3
−−−−−−−→
−1 −2 3
0 −4 4
0 1 −1
−R1→R1
−R2/4→R2
−−−−−−−→
1 2 −3
0 1 −1
0 1 −1
R1−2R2→R1
R3−R2→R3
−−−−−−−→
1 0 −1
0 1 −1
0 0 0
⇒ v1 − v3 = 0 v2 − v3 = 0
⇒ v1 = v2 = v3 = α
An eigenvector corresponding to λ2 = 3 is
1
1
1
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
18. Example
To find the eigenspace corresponding to λ3 = −2, we solve
the homogeneous system (A − λ3I) w = 0.
⇒ (A + 2I) w = 0
⇒
4 −2 3
1 3 1
1 3 1
·
w1
w2
w3
= 0
⇒
4 −2 3
1 3 1
1 3 1
R1−4R2→R1
R3−R2→R3
−−−−−−−→
0 −14 −1
1 3 1
0 0 0
R1↔R2
−−−−→
1 3 1
0 −14 −1
0 0 0
−R2/14→R2
−−−−−−−→
1 3 1
0 1 1/14
0 0 0
R1−3R2→R1
−−−−−−−→
1 0 11/14
0 1 1/14
0 0 0
⇒ w1 + 11w3/14 = 0 w2 + w3/14 = 0
⇒ w1 = 11α w2 = α w3 = −14α
An eigenvector corresponding to λ3 = −2 is
11
1
−14
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization
20. References
Linear Algebra With Applications, 7th Edition
by Steven J. Leon.
Elementary Linear Algebra with Applications, 9th Edition
by Bernard Kolman and David Hill.
http://www.sharetechnote.com/html/Handbook_EngMath_Matrix_
Eigen.html
https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
Dr. Ceni Babaoglu cenibabaoglu.com
Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagonalization