The document provides an overview of computing eigenvalues and eigenvectors. Eigenvalues are the roots of the characteristic polynomial of a matrix A. Eigenvectors corresponding to an eigenvalue λ are non-zero vectors that satisfy the equation (A - λI)v = 0. Examples are provided to demonstrate finding the eigenvalues and eigenvectors of 2x2 matrices by solving the characteristic polynomial and resulting systems of equations. The eigenspace corresponding to an eigenvalue is the span of its eigenvectors.
This document summarizes various tests that can be used to determine if an infinite series converges or diverges, including:
1) The divergence test, integral test, p-series test, comparison test, limit comparison test, alternating series test, and ratio test.
2) It also discusses power series, including determining the radius of convergence and using Taylor series approximations with Taylor's inequality to estimate the remainder term.
3) Key concepts are that convergence tests check if partial sums approach a limit, while divergence tests examine the behavior of individual terms, and that power series have a radius of convergence determining the interval on which they converge.
This document provides an overview of the Laplace transform and its properties and applications. Specifically, it defines the Laplace transform and inverse Laplace transform, lists several Laplace transform pairs, and provides examples of using the Laplace transform to solve initial value problems involving differential equations with constant coefficients. It also includes a problem set with exercises calculating Laplace transforms and using them to solve initial value problems.
Ian.petrow【transcendental number theory】.Tong Leung
This document provides an introduction and overview of the course "Math 249A Fall 2010: Transcendental Number Theory" taught by Kannan Soundararajan. It discusses topics that will be covered, including proving that specific numbers like e, π, and combinations of them are transcendental. Theorems are presented on approximating algebraic numbers and showing linear independence of exponential functions of algebraic numbers. Examples are given of using an integral technique to derive contradictions and prove transcendence.
Peer instructions questions for basic quantum mechanicsmolmodbasics
The document discusses the development of quantum mechanics from Planck/Einstein's quantization of energy to Schrodinger's wave equation. It presents the time-dependent and time-independent Schrodinger equations and their application to particles in a box, harmonic oscillators, and the hydrogen atom. The hydrogen atom energy levels and wavefunctions of the 1s and 2s orbitals are shown.
1) A parabola is the locus of a point whose distance from a fixed point (focus) is equal to its distance from a fixed line (directrix).
2) The standard form equations of a parabola are y2 = 4ax for a vertical axis parabola and x2 = 4ay for a horizontal axis parabola.
3) To sketch a parabola, one must identify the vertex, focus, and direction of opening based on the standard or non-standard form equation and draw the parabolic shape.
This document discusses techniques for evaluating integrals involving exponential functions. It introduces the formulas for integrating exponentials and differentiating them. Several important definite integrals are evaluated, such as the integral from 0 to infinity of e^-ax dx = 1/a. Graphs are used to visualize these integrals. The document then evaluates the more complex integral from negative infinity to positive infinity of e^-ax^2 dx using a change of variables technique. Finally, it discusses how these integrals can be used in kinetic theory and derives an important ratio and normalization factor for Maxwell's velocity distribution.
1. There will be a quiz on Quiz 4 after the next lecture. Exam 2 will be on Feb 25 and cover material from Exam 1 to what is covered on Feb 22.
2. A practice exam will be uploaded on Feb 22 after the remaining material is covered. Optional topics on Feb 23 will not be covered on the exam.
3. Review session on Feb 24 in class. Office hours on Feb 24 from 1-4pm.
11.[95 103]solution of telegraph equation by modified of double sumudu transf...Alexander Decker
1. The document presents a new mathematical transform called the double Elzaki transform.
2. This transform is used to solve the general linear telegraph equation, which is an important partial differential equation in physics.
3. The key steps are: taking the double Elzaki transform of the telegraph equation, taking the single Elzaki transform of the boundary and initial conditions, substituting these into the transformed equation, and taking the inverse transforms to obtain the solution.
This document summarizes various tests that can be used to determine if an infinite series converges or diverges, including:
1) The divergence test, integral test, p-series test, comparison test, limit comparison test, alternating series test, and ratio test.
2) It also discusses power series, including determining the radius of convergence and using Taylor series approximations with Taylor's inequality to estimate the remainder term.
3) Key concepts are that convergence tests check if partial sums approach a limit, while divergence tests examine the behavior of individual terms, and that power series have a radius of convergence determining the interval on which they converge.
This document provides an overview of the Laplace transform and its properties and applications. Specifically, it defines the Laplace transform and inverse Laplace transform, lists several Laplace transform pairs, and provides examples of using the Laplace transform to solve initial value problems involving differential equations with constant coefficients. It also includes a problem set with exercises calculating Laplace transforms and using them to solve initial value problems.
Ian.petrow【transcendental number theory】.Tong Leung
This document provides an introduction and overview of the course "Math 249A Fall 2010: Transcendental Number Theory" taught by Kannan Soundararajan. It discusses topics that will be covered, including proving that specific numbers like e, π, and combinations of them are transcendental. Theorems are presented on approximating algebraic numbers and showing linear independence of exponential functions of algebraic numbers. Examples are given of using an integral technique to derive contradictions and prove transcendence.
Peer instructions questions for basic quantum mechanicsmolmodbasics
The document discusses the development of quantum mechanics from Planck/Einstein's quantization of energy to Schrodinger's wave equation. It presents the time-dependent and time-independent Schrodinger equations and their application to particles in a box, harmonic oscillators, and the hydrogen atom. The hydrogen atom energy levels and wavefunctions of the 1s and 2s orbitals are shown.
1) A parabola is the locus of a point whose distance from a fixed point (focus) is equal to its distance from a fixed line (directrix).
2) The standard form equations of a parabola are y2 = 4ax for a vertical axis parabola and x2 = 4ay for a horizontal axis parabola.
3) To sketch a parabola, one must identify the vertex, focus, and direction of opening based on the standard or non-standard form equation and draw the parabolic shape.
This document discusses techniques for evaluating integrals involving exponential functions. It introduces the formulas for integrating exponentials and differentiating them. Several important definite integrals are evaluated, such as the integral from 0 to infinity of e^-ax dx = 1/a. Graphs are used to visualize these integrals. The document then evaluates the more complex integral from negative infinity to positive infinity of e^-ax^2 dx using a change of variables technique. Finally, it discusses how these integrals can be used in kinetic theory and derives an important ratio and normalization factor for Maxwell's velocity distribution.
1. There will be a quiz on Quiz 4 after the next lecture. Exam 2 will be on Feb 25 and cover material from Exam 1 to what is covered on Feb 22.
2. A practice exam will be uploaded on Feb 22 after the remaining material is covered. Optional topics on Feb 23 will not be covered on the exam.
3. Review session on Feb 24 in class. Office hours on Feb 24 from 1-4pm.
11.[95 103]solution of telegraph equation by modified of double sumudu transf...Alexander Decker
1. The document presents a new mathematical transform called the double Elzaki transform.
2. This transform is used to solve the general linear telegraph equation, which is an important partial differential equation in physics.
3. The key steps are: taking the double Elzaki transform of the telegraph equation, taking the single Elzaki transform of the boundary and initial conditions, substituting these into the transformed equation, and taking the inverse transforms to obtain the solution.
The document provides an overview of kinks and defects in the sine-Gordon equation. It discusses kink solutions that interpolate between potential vacua and can be boosted. While there are no static multi-soliton solutions, breathers represent a bound kink-antikink state. Adding a defect term breaks translational invariance, preventing analytical solutions. Numerical solutions show a kink scattering off a defect within a certain region. An ansatz is presented for modeling kink scattering.
1) Laplace's equation describes situations where the electric potential (V) or other scalar field satisfies ∇^2V = 0. It can be solved in one, two, or three dimensions using separation of variables.
2) In three dimensions, the general solution is a sum of multipole terms involving associated Legendre polynomials. The leading terms are the monopole and dipole contributions.
3) For a dipole potential, the electric field is proportional to p/r^3 where p is the dipole moment. The field points radially away from a head-to-tail dipole and has no φ dependence.
The document discusses the Fourier series representation of periodic functions with an arbitrary period. It provides the general form of the Fourier series for a function f(x) with period 2L, defined over the interval c < x < c + 2L. It also gives the specific forms when c = 0, -L, or L. An example of finding the Fourier series of the function f(x) = x^2 from 0 to 2 is worked out step-by-step.
This document defines and discusses open and closed sets in topology. It begins by precisely defining open and closed sets, noting that a set is open if every point is an interior point, and closed if its complement is open. Examples are provided of open, closed, and neither open nor closed sets. The key properties of unions and intersections of open and closed sets are proven. The closure of a set is then defined as the set of all points close to the original set, where close means every neighborhood of the point intersects the set. Examples of closures of sets are given. Important facts about closures are stated, including that the closure of a set contains the set, and the closure of a subset is contained within
The document summarizes the complex form of Fourier series. It states that after substituting sine and cosine terms into the Fourier series formula, the complex form involves a summation of terms with coefficients multiplied by exponential terms with integer multiples of i and x. It provides the formulas for calculating the coefficients c0, c1, c2, etc. and gives an example function defined over an interval to demonstrate the complex form.
This document discusses rank-aware thresholding algorithms for compressed sensing. It begins by introducing compressed sensing and explaining how traditional linear algebra techniques cannot be used to recover sparse signals from undersampled measurements. It then describes how thresholding and rank-aware thresholding algorithms work by exploiting the sparsity of signals. The key points are that rank-aware thresholding outperforms standard thresholding by eliminating the "square-root bottleneck" and requires only O(k) measurements, versus O(k^2) for thresholding. Simulation results demonstrate this improvement. The document concludes by discussing modeling techniques to predict algorithm performance on very large problems that are impractical to simulate directly.
The document summarizes the RSA cryptosystem. It uses exponentiation modulo n as a one-way function to encrypt and decrypt messages. Specifically:
1. Two large prime numbers p and q are used to compute n = pq.
2. A public exponent e and private exponent d are chosen such that ed = 1 mod φ(n), where φ(n) is the Euler totient function.
3. Encryption is computing the ciphertext as c = me mod n for a message m. Decryption is computing the plaintext as m = cd mod n.
The security of RSA relies on the assumption that factoring large numbers n is computationally difficult. A proof of security would
Essay Mojo is one of the most reliable writing services that cover all the disciplines of academic writing. It assures high quality content after conducting in-depth research on the prospective subject at the most reasonable prices.
The document provides a user's guide for the HP DesignJet 2500CP/2000CP printer, covering topics such as using the front panel, working with media, controlling print quality, and maintaining the printer. It includes information on loading and unloading media, replacing ink systems, nesting pages to reduce waste, and troubleshooting image quality issues. The guide also details how to view and change settings through the front panel such as page size, margins, and print quality.
This document provides troubleshooting information for the HP DesignJet 2500CP and 2000CP printers. It lists common error codes and their potential causes and solutions. The document also provides tips on issues like print quality problems, component failures, and calibration errors. Technicians are advised to follow the steps provided to methodically diagnose and resolve issues.
La disciplina de la libertad - Fernando SavaterAvy Aguilar
El documento discute cómo la educación y la enseñanza son necesarias para lograr la autonomía y libertad. Aunque la enseñanza implica cierta coacción, ayuda a los niños a adquirir las herramientas simbólicas necesarias para explorar el mundo por su cuenta. El objetivo de la educación es formar seres humanos mediante una "obra de arte colectiva" que moldea a los individuos para que sean libres y autónomos.
Este documento narra la trayectoria profesional del autor desde sus inicios trabajando en el negocio familiar hasta convertirse en un experto vendedor. Comienza describiendo sus primeras experiencias atendiendo un mostrador y luego negociando con proveedores. Más adelante consigue su primer empleo formal como vendedor de productos químicos, donde aprende sobre estrategias básicas de ventas como realizar visitas a clientes y establecer pronósticos de ventas. El autor reflexiona sobre cómo sus diferentes experiencias lo han ayudado a desar
La empresa ofrece productos, servicios, asesorías y herramientas para el comercio e industria, con el objetivo de promover el desarrollo del talento humano en las organizaciones. Busca satisfacer las necesidades de desarrollo de sus clientes a través de soluciones adecuadas que fomenten el crecimiento económico y la sinergia entre equipos de trabajo.
Este documento proporciona instrucciones sobre cómo presentar una monografía. Explica que una monografía es un estudio sistemático y profundo de un tema específico. Luego enumera los 8 pasos para presentar una monografía: 1) Hoja de respeto, 2) Carátula, 3) Índice, 4) Introducción, 5) Cuerpo o contenido, 6) Conclusiones, 7) Bibliografía, 8) Anexos. Proporciona detalles sobre cada sección y los formatos y estilos requeridos.
La empresa ofrece una variedad de productos y servicios de asesoría para el desarrollo empresarial y del talento humano en organizaciones internacionales, incluyendo asesoría empresarial, capacitación, desarrollo de equipos de trabajo, comercio internacional, y evaluación y optimización de procesos. Su objetivo es satisfacer las necesidades de desarrollo de sus clientes a través de soluciones efectivas que promuevan el crecimiento económico.
El documento proporciona instrucciones en 7 pasos para preparar una monografía, incluyendo la selección de un tema, búsqueda y evaluación de información, plan de trabajo, redacción, bibliografía y estilo APA. También incluye un ejemplo de plan de trabajo con portada, introducción, cuerpo, conclusión y bibliografía.
This document provides 18 problems related to matrices. The problems cover topics like finding eigen values and vectors, properties of eigen values under operations like inverse and powers, and applying Cayley-Hamilton theorem.
The document discusses eigenvalues and eigenvectors of linear transformations and matrices. It begins by defining a diagonalizable matrix as one that can be transformed into a diagonal matrix through a change of basis. It then defines eigenvalues and eigenvectors for both linear transformations and matrices. The characteristic polynomial of a matrix is introduced, which has roots that are the eigenvalues of the matrix. It is shown that the algebraic multiplicity of an eigenvalue is equal to its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace. The algebraic multiplicity is always greater than or equal to the geometric multiplicity.
This document discusses sequences and series of numbers. It begins by introducing sequences and the concept of convergence for sequences. A sequence converges to a limit L if, given any positive number ε, there exists an N such that the terms an of the sequence satisfy |L - an| < ε for all n > N.
It then proves some basic properties of convergence, including that a convergent sequence must be bounded, and that limits are preserved under operations. Cauchy's criterion for convergence is introduced - a sequence is Cauchy if given any ε, there exists an N such that |am - an| < ε for all m, n > N. Every convergent sequence is Cauchy, and C
The document discusses Legendre polynomials, which are special functions that arise in solutions to Laplace's equation in spherical coordinates. Some key points:
1) Legendre polynomials Pn(cosθ) are a set of orthogonal polynomials that satisfy Legendre's differential equation.
2) Pn(cosθ) can be defined using a generating function or by taking partial derivatives of 1/r.
3) Important properties of Legendre polynomials include P0(t)=1, Pn(1)=1, Pn(-1)=(-1)n, and a recurrence relation involving Pn+1, Pn, and their derivatives.
1. A complex number λ is an eigenvalue of a matrix A if there exists a non-zero vector x such that Ax = λx.
2. If a matrix has complex eigenvalues, it provides important information about the matrix, such as in problems involving vibrations and rotations in space.
3. For a complex eigenvalue λ = a + bi, a is called the real part and b is called the imaginary part. The absolute value |λ| represents the "length" or magnitude of the eigenvalue.
The document provides an overview of kinks and defects in the sine-Gordon equation. It discusses kink solutions that interpolate between potential vacua and can be boosted. While there are no static multi-soliton solutions, breathers represent a bound kink-antikink state. Adding a defect term breaks translational invariance, preventing analytical solutions. Numerical solutions show a kink scattering off a defect within a certain region. An ansatz is presented for modeling kink scattering.
1) Laplace's equation describes situations where the electric potential (V) or other scalar field satisfies ∇^2V = 0. It can be solved in one, two, or three dimensions using separation of variables.
2) In three dimensions, the general solution is a sum of multipole terms involving associated Legendre polynomials. The leading terms are the monopole and dipole contributions.
3) For a dipole potential, the electric field is proportional to p/r^3 where p is the dipole moment. The field points radially away from a head-to-tail dipole and has no φ dependence.
The document discusses the Fourier series representation of periodic functions with an arbitrary period. It provides the general form of the Fourier series for a function f(x) with period 2L, defined over the interval c < x < c + 2L. It also gives the specific forms when c = 0, -L, or L. An example of finding the Fourier series of the function f(x) = x^2 from 0 to 2 is worked out step-by-step.
This document defines and discusses open and closed sets in topology. It begins by precisely defining open and closed sets, noting that a set is open if every point is an interior point, and closed if its complement is open. Examples are provided of open, closed, and neither open nor closed sets. The key properties of unions and intersections of open and closed sets are proven. The closure of a set is then defined as the set of all points close to the original set, where close means every neighborhood of the point intersects the set. Examples of closures of sets are given. Important facts about closures are stated, including that the closure of a set contains the set, and the closure of a subset is contained within
The document summarizes the complex form of Fourier series. It states that after substituting sine and cosine terms into the Fourier series formula, the complex form involves a summation of terms with coefficients multiplied by exponential terms with integer multiples of i and x. It provides the formulas for calculating the coefficients c0, c1, c2, etc. and gives an example function defined over an interval to demonstrate the complex form.
This document discusses rank-aware thresholding algorithms for compressed sensing. It begins by introducing compressed sensing and explaining how traditional linear algebra techniques cannot be used to recover sparse signals from undersampled measurements. It then describes how thresholding and rank-aware thresholding algorithms work by exploiting the sparsity of signals. The key points are that rank-aware thresholding outperforms standard thresholding by eliminating the "square-root bottleneck" and requires only O(k) measurements, versus O(k^2) for thresholding. Simulation results demonstrate this improvement. The document concludes by discussing modeling techniques to predict algorithm performance on very large problems that are impractical to simulate directly.
The document summarizes the RSA cryptosystem. It uses exponentiation modulo n as a one-way function to encrypt and decrypt messages. Specifically:
1. Two large prime numbers p and q are used to compute n = pq.
2. A public exponent e and private exponent d are chosen such that ed = 1 mod φ(n), where φ(n) is the Euler totient function.
3. Encryption is computing the ciphertext as c = me mod n for a message m. Decryption is computing the plaintext as m = cd mod n.
The security of RSA relies on the assumption that factoring large numbers n is computationally difficult. A proof of security would
Essay Mojo is one of the most reliable writing services that cover all the disciplines of academic writing. It assures high quality content after conducting in-depth research on the prospective subject at the most reasonable prices.
The document provides a user's guide for the HP DesignJet 2500CP/2000CP printer, covering topics such as using the front panel, working with media, controlling print quality, and maintaining the printer. It includes information on loading and unloading media, replacing ink systems, nesting pages to reduce waste, and troubleshooting image quality issues. The guide also details how to view and change settings through the front panel such as page size, margins, and print quality.
This document provides troubleshooting information for the HP DesignJet 2500CP and 2000CP printers. It lists common error codes and their potential causes and solutions. The document also provides tips on issues like print quality problems, component failures, and calibration errors. Technicians are advised to follow the steps provided to methodically diagnose and resolve issues.
La disciplina de la libertad - Fernando SavaterAvy Aguilar
El documento discute cómo la educación y la enseñanza son necesarias para lograr la autonomía y libertad. Aunque la enseñanza implica cierta coacción, ayuda a los niños a adquirir las herramientas simbólicas necesarias para explorar el mundo por su cuenta. El objetivo de la educación es formar seres humanos mediante una "obra de arte colectiva" que moldea a los individuos para que sean libres y autónomos.
Este documento narra la trayectoria profesional del autor desde sus inicios trabajando en el negocio familiar hasta convertirse en un experto vendedor. Comienza describiendo sus primeras experiencias atendiendo un mostrador y luego negociando con proveedores. Más adelante consigue su primer empleo formal como vendedor de productos químicos, donde aprende sobre estrategias básicas de ventas como realizar visitas a clientes y establecer pronósticos de ventas. El autor reflexiona sobre cómo sus diferentes experiencias lo han ayudado a desar
La empresa ofrece productos, servicios, asesorías y herramientas para el comercio e industria, con el objetivo de promover el desarrollo del talento humano en las organizaciones. Busca satisfacer las necesidades de desarrollo de sus clientes a través de soluciones adecuadas que fomenten el crecimiento económico y la sinergia entre equipos de trabajo.
Este documento proporciona instrucciones sobre cómo presentar una monografía. Explica que una monografía es un estudio sistemático y profundo de un tema específico. Luego enumera los 8 pasos para presentar una monografía: 1) Hoja de respeto, 2) Carátula, 3) Índice, 4) Introducción, 5) Cuerpo o contenido, 6) Conclusiones, 7) Bibliografía, 8) Anexos. Proporciona detalles sobre cada sección y los formatos y estilos requeridos.
La empresa ofrece una variedad de productos y servicios de asesoría para el desarrollo empresarial y del talento humano en organizaciones internacionales, incluyendo asesoría empresarial, capacitación, desarrollo de equipos de trabajo, comercio internacional, y evaluación y optimización de procesos. Su objetivo es satisfacer las necesidades de desarrollo de sus clientes a través de soluciones efectivas que promuevan el crecimiento económico.
El documento proporciona instrucciones en 7 pasos para preparar una monografía, incluyendo la selección de un tema, búsqueda y evaluación de información, plan de trabajo, redacción, bibliografía y estilo APA. También incluye un ejemplo de plan de trabajo con portada, introducción, cuerpo, conclusión y bibliografía.
This document provides 18 problems related to matrices. The problems cover topics like finding eigen values and vectors, properties of eigen values under operations like inverse and powers, and applying Cayley-Hamilton theorem.
The document discusses eigenvalues and eigenvectors of linear transformations and matrices. It begins by defining a diagonalizable matrix as one that can be transformed into a diagonal matrix through a change of basis. It then defines eigenvalues and eigenvectors for both linear transformations and matrices. The characteristic polynomial of a matrix is introduced, which has roots that are the eigenvalues of the matrix. It is shown that the algebraic multiplicity of an eigenvalue is equal to its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace. The algebraic multiplicity is always greater than or equal to the geometric multiplicity.
This document discusses sequences and series of numbers. It begins by introducing sequences and the concept of convergence for sequences. A sequence converges to a limit L if, given any positive number ε, there exists an N such that the terms an of the sequence satisfy |L - an| < ε for all n > N.
It then proves some basic properties of convergence, including that a convergent sequence must be bounded, and that limits are preserved under operations. Cauchy's criterion for convergence is introduced - a sequence is Cauchy if given any ε, there exists an N such that |am - an| < ε for all m, n > N. Every convergent sequence is Cauchy, and C
The document discusses Legendre polynomials, which are special functions that arise in solutions to Laplace's equation in spherical coordinates. Some key points:
1) Legendre polynomials Pn(cosθ) are a set of orthogonal polynomials that satisfy Legendre's differential equation.
2) Pn(cosθ) can be defined using a generating function or by taking partial derivatives of 1/r.
3) Important properties of Legendre polynomials include P0(t)=1, Pn(1)=1, Pn(-1)=(-1)n, and a recurrence relation involving Pn+1, Pn, and their derivatives.
1. A complex number λ is an eigenvalue of a matrix A if there exists a non-zero vector x such that Ax = λx.
2. If a matrix has complex eigenvalues, it provides important information about the matrix, such as in problems involving vibrations and rotations in space.
3. For a complex eigenvalue λ = a + bi, a is called the real part and b is called the imaginary part. The absolute value |λ| represents the "length" or magnitude of the eigenvalue.
Partial midterm set7 soln linear algebrameezanchand
This document provides solutions to problems from Problem Set 7 in 18.06 Linear Algebra. It includes solutions to 6 problems involving eigenvalues and eigenvectors of matrices. Key details include:
- Finding eigenvalues and eigenvectors of specific matrices like A = [matrix]
- Showing that the characteristic polynomial of a matrix A equals 0 using its diagonalization
- Deriving that the inverse of an invertible matrix A can be written as a polynomial function of A
- Explaining that the eigenvalues of a matrix A are also the eigenvalues of its transpose AT, while the eigenvectors may differ.
The document discusses the expected value and variance of a Poisson random variable X with parameter λ. It is shown that:
1) The expected value E(X) = λ through calculating the summation from x=0 to infinity of x*e^-λ*(λ^x)/x!.
2) The variance Var(X) = λ through calculating E(X^2) - E(X)^2 and showing E(X^2) = λ^2 and E(X) = λ.
3) In summary, both the expected value and variance of a Poisson random variable X are equal to its parameter λ.
1) An eigenvector of a square matrix A is a non-zero vector x that satisfies the equation Ax = λx, where λ is the corresponding eigenvalue.
2) The zero vector cannot be an eigenvector, but λ = 0 can be an eigenvalue.
3) For a matrix A, the eigenvectors and eigenvalues can be found by solving the system of equations (A - λI)x = 0, where λI is the identity matrix multiplied by the eigenvalue λ.
Lesson 24: Area and Distances (worksheet solutions)Matthew Leingang
This document provides solutions to a calculus worksheet involving estimating the area under the curve of the function f(x) = ex on the interval [0,1].
[1] The solution estimates the area at increasing levels of subdivision (n = 2, 4, 8) and finds the area is approaching a limit.
[2] Closed form expressions are derived for the left and right approximations Ln and Rn to the area, showing Rn = e1/nLn.
[3] The common limit of Ln and Rn as n approaches infinity is shown to be 1-e, which is the exact area under the curve on the interval.
This document summarizes Markowitz's mean-variance portfolio theory and the two-fund theorem.
[1] Markowitz formulated the mean-variance model, which minimizes portfolio variance subject to a target expected return. The optimal weights are a function of the covariance matrix and target mean.
[2] The two-fund theorem states that any efficient portfolio can be replicated as a combination of two "fundamental" portfolios. Investors only need to invest in these two funds.
[3] The minimum variance set forms the left boundary of the feasible region in mean-variance space. Portfolios on this boundary are efficient funds.
This document contains information about positive definite matrices and eigenvectors/eigenvalues. It provides an example of a positive definite matrix with all positive eigenvalues. It also gives an example of showing that the transpose of a matrix A multiplied by itself (ATA) is positive definite if A is invertible. Finally, it provides an example of finding the eigenvectors and eigenvalues of a 2x2 matrix. The eigenvectors are solutions to Av=λv, where λ is the eigenvalue.
The document contains announcements about an exam, practice exam, review sessions, and exam grading for a class. It states that Exam 2 will be on Thursday, February 25 in class. A practice exam will be uploaded by 2 pm that day. Optional review topics will be covered the next day but will not be on the exam. A review session will be held on Wednesday with office hours from 1-4 pm. It also reminds students that a different class starts on Monday and to collect graded exams on Friday between 7 am and 6 pm.
This document defines and provides examples of sequences of real numbers. It begins by defining a sequence as a set of numbers written in a definite order. Examples are provided to illustrate bounded, increasing, decreasing, convergent, and divergent sequences. The limit of a sequence is defined as the number L that the terms of the sequence approach as n becomes large. A sequence is convergent if its limit exists and divergent otherwise. Bounded sequences are those for which the terms are all less than some positive number M, but bounded sequences may still diverge. Recursively defined sequences are also discussed.
This document provides an overview of algebraic techniques in combinatorics, including linear algebra concepts, partially ordered sets (posets), and examples of problems solved using these techniques. Some key points discussed are:
- Useful linear algebra facts such as rank, determinants, and vector/matrix properties
- Definitions and representations of posets, including Dilworth's theorem relating chains and antichains
- Examples of combinatorial problems solved using linear algebra tools such as vectors/matrices or applying Dilworth's theorem to obtain a divisibility relation poset
Eigenvalues and eigenfunctions are key concepts in linear algebra. An eigenfunction is a function that when operated on by a linear operator produces a constant multiplied version of itself. The constant is the corresponding eigenvalue. Eigenvalues are the solutions to the characteristic polynomial of the linear operator. Eigenfunctions are not unique as any constant multiple of an eigenfunction is also an eigenfunction with the same eigenvalue. The spectrum of an operator is the set of all its eigenvalues.
The document shows mathematical operations to solve a system of equations. It starts with two equations relating x, y, λ. It then takes the inverse of a matrix and performs multiplication. This results in another equation relating x and y in terms of λ. Simplifying this equation gives the solution of x = 1 and y = -λ.
Change of variables in double integralsTarun Gehlot
1. The document discusses change of variables for double integrals, introducing the Jacobian determinant which relates the differentials of the original and transformed variables.
2. It provides an example of using a change of variables (u=x-y, v=x+y) to evaluate an integral over a parallelogram region.
3. Polar coordinates are also discussed as a common change of variables technique for double integrals, with an example evaluating an integral over a circular region in polar coordinates.
Change of variables in double integralsTarun Gehlot
1. The document discusses change of variables for double integrals, introducing the Jacobian determinant which relates the differentials of the original and transformed variables.
2. It provides an example of using a change of variables (u=x-y, v=x+y) to evaluate an integral over a parallelogram region.
3. Polar coordinates are also discussed as a common change of variables technique for double integrals, with an example evaluating an integral over a circular region in polar coordinates.
The document discusses infinite series and sequences. It begins by introducing the concept of an infinite sum and examines whether a sum like 1/2 + 1/4 + 1/8 + ... can be assigned a numerical value. It then defines an infinite series as a sum that continues indefinitely, and a sequence as the individual terms in a series. The key points are:
- An infinite series can be assigned a value by taking the limit of the corresponding sequence of partial sums.
- Common examples like 0.333... and π are actually infinite series.
- Sequences are functions with domain the natural numbers, and their limit is defined similarly to limits of functions.
- Monotonic and bounded sequences are important for
1. Harvey Mudd College Math Tutorial:
Eigenvalues and Eigenvectors
We review here the basics of computing eigenvalues and eigenvectors. Eigenvalues and
eigenvectors play a prominent role in the study of ordinary differential equations and in
many applications in the physical sciences. Expect to see them come up in a variety of
contexts!
Definitions
Let A be an n × n matrix. The number
λ is an eigenvalue of A if there exists a
non-zero vector v such that
Av = λv.
In this case, vector v is called an eigen-
vector of A corresponding to λ.
Computing Eigenvalues and Eigenvectors
We can rewrite the condition Av = λv as
(A − λI)v = 0.
Otherwise, if A − λI has an inverse,
where I is the n×n identity matrix. Now, in order
for a non-zero vector v to satisfy this equation, (A − λI)−1 (A − λI)v = (A − λI)−1 0
A − λI must not be invertible. v = 0.
That is, the determinant of A − λI must equal 0.
We call p(λ) = det(A − λI) the characteristic But we are looking for a non-zero vector v.
polynomial of A. The eigenvalues of A are sim-
ply the roots of the characteristic polynomial of
A.
Example
2 −4 2−λ −4
Let A = . Then p(λ) = det
−1 −1 −1 −1 − λ
= (2 − λ)(−1 − λ) − (−4)(−1)
= λ2 − λ − 6
= (λ − 3)(λ + 2).
2. Thus, λ1 = 3 and λ2 = −2 are the eigenvalues of A.
v1
v2
To find eigenvectors v = .
.
corresponding to an eigenvalue λ, we simply solve the system
.
vn
of linear equations given by
(A − λI)v = 0.
Example
2 −4
The matrix A = of the previous example has eigenvalues λ1 = 3 and λ2 = −2.
−1 −1
Let’s find the eigenvectors corresponding to λ1 = 3. Let v = v1 . Then (A − 3I)v = 0 gives
v2
us
2−3 −4 v1 0
= ,
−1 −1 − 3 v2 0
from which we obtain the duplicate equations
−v1 − 4v2 = 0
−v1 − 4v2 = 0.
If we let v2 = t, then v1 = −4t. All eigenvectors corresponding to λ1 = 3 are multiples of
−4
1
and thus the eigenspace corresponding to λ1 = 3 is given by the span of −4 . That is,
1
−4
1
is a basis of the eigenspace corresponding to λ1 = 3.
Repeating this process with λ2 = −2, we find that
4v1 − 4V2 = 0
−v1 + v2 = 0
1
If we let v2 = t then v1 = t as well. Thus, an eigenvector corresponding to λ2 = −2 is 1
1 1
and the eigenspace corresponding to λ2 = −2 is given by the span of 1
. 1
is a basis
for the eigenspace corresponding to λ2 = −2.
In the following example, we see a two-dimensional eigenspace.
Example
5 8 16 5−λ 8 16
Let A = 4 1 8 . Then p(λ) = det 4 1−λ 8 = (λ − 1)(λ + 3)2
−4 −4 −11 −4 −4 −11 − λ
aftersome algebra! Thus, λ1 = 1 and λ2 = −3 are the eigenvalues of A. Eigenvectors
v1
v = v2 corresponding to λ1 = 1 must satisfy
v3
3. 4v1 + 8v2 + 16v3 = 0
4v1 + 8v3 = 0
−4v1 − 4v2 − 12v3 = 0.
Letting v3 = t, we find from the second equation that v1 = −2t, and then v2 = −t. All
−2
eigenvectors corresponding to λ1 = 1 are multiples of −1 , and so the eigenspace corre-
1
−2 −2
sponding to λ1 = 1 is given by the span of −1 . −1 is a basis for the eigenspace
1
1
corresponding to λ1 = 1.
Eigenvectors corresponding to λ2 = −3 must satisfy
8v1 + 8v2 + 16v3 = 0
4v1 + 4v2 + 8v3 = 0
−4v1 − 4v2 − 8v3 = 0.
The equations here are just multiples of each other! If we let v3 = t and v2 = s, then
v1 = −s − 2t. Eigenvectors corresponding to λ2 = −3 have the form
−1 −2
1 s + 0 t.
0 1
−1
Thus, the eigenspace corresponding to λ2 = −3 is two-dimensional and is spanned by 1
0
−2 −1
−2
and 0 . 1 , 0 is a basis for the eigenspace corresponding to λ2 = −3.
1
0 1
Notes
• Eigenvalues and eigenvectors can be complex-valued as well as real-valued.
• The dimension of the eigenspace corresponding to an eigenvalue is less than or equal
to the multiplicity of that eigenvalue.
• The techniques used here are practical for 2 × 2 and 3 × 3 matrices. Eigenvalues and
eigenvectors of larger matrices are often found using other techniques, such as iterative
methods.
4. Key Concepts
Let A be an n×n matrix. The eigenvalues of A are the roots of the characteristic polynomial
p(λ) = det(A − λI).
v1
v2
For each eigenvalue λ, we find eigenvectors v = .
.
by solving the linear system
.
vn
(A − λI)v = 0.
The set of all vectors v satisfying Av = λv is called the eigenspace of A corresponding to
λ.
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