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This document provides solutions to problems from Gilbert Strang's Linear Algebra textbook. It derives the decomposition of a matrix A into its basis for the row space and nullspace. It then provides solutions to several problems from Chapter 1 on vectors and Chapter 2 on solving linear equations. The problems cover topics like counting dimensions, vector sums representing hours in a day, and checking properties of subspaces. The document uses notation like ⇒ to represent row reduction steps and derives expressions for matrix inverses and solutions to systems of equations.
This document provides contact information for math assignment help, including a phone number and email address. It then presents solutions to several problems from a linear algebra textbook. The problems cover topics like writing a quadratic form as a sum of squares, finding the closest line and plane of best fit to a set of points, orthonormal vectors, and determinants. Solutions are provided in mathematical notation and include working steps.
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This document provides solutions to problems from Gilbert Strang's Linear Algebra textbook. It derives the decomposition of a matrix A into its basis for the row space and nullspace. It then provides solutions to several problems from Chapter 1 on vectors and Chapter 2 on solving linear equations. The problems cover topics like counting dimensions, vector sums representing hours in a day, and checking properties of subspaces. The document uses notation like ⇒ to represent row reduction steps and derives expressions for matrix inverses and solutions to systems of equations.
This document provides contact information for math assignment help, including a phone number and email address. It then presents solutions to several problems from a linear algebra textbook. The problems cover topics like writing a quadratic form as a sum of squares, finding the closest line and plane of best fit to a set of points, orthonormal vectors, and determinants. Solutions are provided in mathematical notation and include working steps.
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The simplex algorithm is used to solve linear programming problems by iteratively moving from one basic feasible solution (BFS) to another with a better objective value. It starts at an initial BFS and checks if it is optimal. If not, it moves to a new improved BFS and repeats the process until an optimal solution is found. For the example problem of maximizing profit from two drugs given machine time constraints, the algorithm starts at BFS (0,0,9,8) and improves to BFS (4,0,5,0) by increasing the variable with the largest coefficient in the objective function while maintaining feasibility.
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The document provides an overview of linear algebra and matrix theory. It discusses the history and development of matrices, defines key matrix concepts like dimensions and operations, and covers foundational topics like matrix addition, multiplication, inverses, and solving systems of linear equations. The document is intended as an introduction to linear algebra and matrices for students.
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This document provides an overview of graphing linear equations. It defines key terms like solutions, intercepts, and linear models. Examples are given to show how to graph equations by finding intercepts or using a table of points. Horizontal and vertical lines are discussed as special cases of linear equations. The document concludes with an example of using a linear equation to model a real-world situation involving monthly phone costs.
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This document provides an overview of linear models and matrix algebra concepts that are important for economics. It discusses the objectives of using mathematics for economics, including understanding problems by stating the unknown and known variables. The document then covers key topics in linear algebra like the history of matrices, what matrices are, basic matrix operations, and properties of matrix addition and multiplication. It also introduces concepts like the inverse and transpose of a matrix. Finally, it provides an example of how matrices and vectors can represent systems of linear equations used in economic models.
This document provides an introduction to matrix algebra and random vectors. It defines key concepts such as vectors, matrices, matrix operations, and properties of positive definite matrices. Vectors are defined as arrays of real numbers that can be added or multiplied by scalars. Matrices are rectangular arrays of numbers that can be added or multiplied. Positive definite matrices are matrices where the quadratic form is always nonnegative. The eigenvalues and eigenvectors of a symmetric positive definite matrix allow geometric interpretation of distances defined by the matrix.
The document defines matrices and their properties, including symmetric, skew-symmetric, and determinant. It provides examples of solving systems of equations using matrices and their inverses. It also discusses properties of determinants, including properties related to symmetric and skew-symmetric matrices. Inverse trigonometric functions are defined, including their domains, ranges, and relationships between inverse functions using addition and subtraction formulas. Sample problems are provided to solve systems of equations and evaluate determinants.
This document summarizes Chris Swierczewski's general exam presentation on computational applications of Riemann surfaces and Abelian functions. The presentation covered the geometry and algebra of Riemann surfaces, including bases of cycles, holomorphic differentials, and period matrices. Applications discussed include using Riemann theta functions to find periodic solutions to integrable PDEs like the Kadomtsev–Petviashvili equation. The talk also discussed linear matrix representations of algebraic curves and the constructive Schottky problem of realizing a Riemann matrix as the period matrix of a curve.
This document contains solutions to several problems involving vector calculus and partial differential equations.
For problem 1, key points include: deriving an identity involving curl and dot products; showing that curl is self-adjoint under certain boundary conditions where the vector field is parallel to the boundary normal; and explaining how Maxwell's equations with these boundary conditions would produce oscillating electromagnetic wave solutions.
Problem 2 involves solving the eigenproblem for the Laplacian in an annular region using separation of variables. Continuity conditions at the inner and outer radii lead to a transcendental equation determining the eigenvalues.
Problem 3 examines eigenproblems for the Laplacian and curl operators, showing they are self-adjoint and obtaining matrix and finite difference
The document explores different types of conic sections, including ellipses, hyperbolas, circles, and parabolas. It shows how to use completing the square to rewrite general form conic equations into standard form equations for each type. The values of coefficients A, B, C, D, E determine whether the conic is an ellipse, hyperbola, circle, or may degenerate into a line or point. When B is not equal to 0, polar coordinates can be used to graph the conic section.
The document discusses linear functions and matrices. It provides examples of solving linear equations and systems of linear equations. It also covers matrix operations such as addition, subtraction, and multiplication. Matrix inverses and determinants are introduced as tools for solving systems of linear equations. Applications include finding linear models from data points and using matrices to represent and solve simultaneous equations.
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Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
This document contains 30 multiple choice questions from the 1995 IIT JEE mathematics exam. The questions cover topics in complex numbers, trigonometry, functions, vectors, and probability. No answers are provided. Students seeking solutions would need to visit an external website. The document provides an unsolved practice test to help students prepare for the competitive Indian engineering entrance exam.
My paper for Domain Decomposition Conference in Strobl, Austria, 2005Alexander Litvinenko
We did a first step in solving, so-called, skin problem. We developed an efficient H-matrix preconditioner to solve diffusion problem with jumping coefficients
This document introduces graph C*-algebras and related concepts:
- A graph C*-algebra is generated by partial isometries and projections representing a directed graph.
- A Cuntz-Krieger E-family satisfies relations involving these operators to represent a graph as bounded operators on a Hilbert space.
- The graph C*-algebra is then the closed algebra generated by this family of operators.
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The simplex algorithm is used to solve linear programming problems by iteratively moving from one basic feasible solution (BFS) to another with a better objective value. It starts at an initial BFS and checks if it is optimal. If not, it moves to a new improved BFS and repeats the process until an optimal solution is found. For the example problem of maximizing profit from two drugs given machine time constraints, the algorithm starts at BFS (0,0,9,8) and improves to BFS (4,0,5,0) by increasing the variable with the largest coefficient in the objective function while maintaining feasibility.
I am Piers L. I am a Calculus Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From, the University of Adelaide. I have been helping students with their homework for the past 7 years. I solve homework related to Calculus.
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The document provides an overview of linear algebra and matrix theory. It discusses the history and development of matrices, defines key matrix concepts like dimensions and operations, and covers foundational topics like matrix addition, multiplication, inverses, and solving systems of linear equations. The document is intended as an introduction to linear algebra and matrices for students.
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This document provides an overview of graphing linear equations. It defines key terms like solutions, intercepts, and linear models. Examples are given to show how to graph equations by finding intercepts or using a table of points. Horizontal and vertical lines are discussed as special cases of linear equations. The document concludes with an example of using a linear equation to model a real-world situation involving monthly phone costs.
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You can also call on +1 678 648 4277 for any assistance with Differential Equations Assignment.
This document provides an overview of linear models and matrix algebra concepts that are important for economics. It discusses the objectives of using mathematics for economics, including understanding problems by stating the unknown and known variables. The document then covers key topics in linear algebra like the history of matrices, what matrices are, basic matrix operations, and properties of matrix addition and multiplication. It also introduces concepts like the inverse and transpose of a matrix. Finally, it provides an example of how matrices and vectors can represent systems of linear equations used in economic models.
This document provides an introduction to matrix algebra and random vectors. It defines key concepts such as vectors, matrices, matrix operations, and properties of positive definite matrices. Vectors are defined as arrays of real numbers that can be added or multiplied by scalars. Matrices are rectangular arrays of numbers that can be added or multiplied. Positive definite matrices are matrices where the quadratic form is always nonnegative. The eigenvalues and eigenvectors of a symmetric positive definite matrix allow geometric interpretation of distances defined by the matrix.
The document defines matrices and their properties, including symmetric, skew-symmetric, and determinant. It provides examples of solving systems of equations using matrices and their inverses. It also discusses properties of determinants, including properties related to symmetric and skew-symmetric matrices. Inverse trigonometric functions are defined, including their domains, ranges, and relationships between inverse functions using addition and subtraction formulas. Sample problems are provided to solve systems of equations and evaluate determinants.
This document summarizes Chris Swierczewski's general exam presentation on computational applications of Riemann surfaces and Abelian functions. The presentation covered the geometry and algebra of Riemann surfaces, including bases of cycles, holomorphic differentials, and period matrices. Applications discussed include using Riemann theta functions to find periodic solutions to integrable PDEs like the Kadomtsev–Petviashvili equation. The talk also discussed linear matrix representations of algebraic curves and the constructive Schottky problem of realizing a Riemann matrix as the period matrix of a curve.
This document contains solutions to several problems involving vector calculus and partial differential equations.
For problem 1, key points include: deriving an identity involving curl and dot products; showing that curl is self-adjoint under certain boundary conditions where the vector field is parallel to the boundary normal; and explaining how Maxwell's equations with these boundary conditions would produce oscillating electromagnetic wave solutions.
Problem 2 involves solving the eigenproblem for the Laplacian in an annular region using separation of variables. Continuity conditions at the inner and outer radii lead to a transcendental equation determining the eigenvalues.
Problem 3 examines eigenproblems for the Laplacian and curl operators, showing they are self-adjoint and obtaining matrix and finite difference
The document explores different types of conic sections, including ellipses, hyperbolas, circles, and parabolas. It shows how to use completing the square to rewrite general form conic equations into standard form equations for each type. The values of coefficients A, B, C, D, E determine whether the conic is an ellipse, hyperbola, circle, or may degenerate into a line or point. When B is not equal to 0, polar coordinates can be used to graph the conic section.
The document discusses linear functions and matrices. It provides examples of solving linear equations and systems of linear equations. It also covers matrix operations such as addition, subtraction, and multiplication. Matrix inverses and determinants are introduced as tools for solving systems of linear equations. Applications include finding linear models from data points and using matrices to represent and solve simultaneous equations.
Are you in search of trustworthy and affordable assistance for your Linear Algebra assignments? Look no further! Our team of experienced math experts specializing in Linear Algebra is here to provide you with reliable help. Whether your assignment is big or small, we offer personalized assistance to ensure that all work is completed to the highest standard. Our prices are competitive and tailored to be student-friendly, making it easier for you to access the help you need.
Why struggle with your Linear Algebra assignments when you can receive expert assistance today? Don't hesitate to contact us now for more information on how our team can support you with your math assignments, specifically in the field of Linear Algebra. Visit our website at www.mathsassignmenthelp.com,
send us an email at info@mathsassignmenthelp.com,
or reach out to us via WhatsApp at +1(315)557-6473.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
This document contains 30 multiple choice questions from the 1995 IIT JEE mathematics exam. The questions cover topics in complex numbers, trigonometry, functions, vectors, and probability. No answers are provided. Students seeking solutions would need to visit an external website. The document provides an unsolved practice test to help students prepare for the competitive Indian engineering entrance exam.
My paper for Domain Decomposition Conference in Strobl, Austria, 2005Alexander Litvinenko
We did a first step in solving, so-called, skin problem. We developed an efficient H-matrix preconditioner to solve diffusion problem with jumping coefficients
This document introduces graph C*-algebras and related concepts:
- A graph C*-algebra is generated by partial isometries and projections representing a directed graph.
- A Cuntz-Krieger E-family satisfies relations involving these operators to represent a graph as bounded operators on a Hilbert space.
- The graph C*-algebra is then the closed algebra generated by this family of operators.
I am Frank Dennis. I love exploring new topics. Academic writing seemed an interesting option for me. After working for many years with mathhomeworksolver.com. I have assisted many students with their Assignments. I can proudly say, each student I have served is happy with the quality of the solution that I have provided. I have acquired my PhD. in Maths, Leeds University UK.
My name is Eric Brian. I have been associated with mathhomeworksolver.com for the past 10 years and have been helping Mathematics students with their Linear Algebra Communications Assignments.
I have a master's in mathematics from the University of Melbourne, Australia.
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This document discusses three problems related to partial differential equations:
1) Finding eigenfunctions and eigenvalues for an operator and bounding the maximum value of a Rayleigh quotient.
2) Solving the Laplacian eigenproblem in a cylinder using finite differences and comparing to analytical solutions.
3) Solving the wave equation for a vibrating string driven by an oscillating force and deriving the Green's function.
This document provides solutions to problems involving complex analysis concepts such as complex arithmetic, functions, mappings, analytic functions, line integrals, and the Cauchy integral formula. It includes step-by-step workings for calculating real and imaginary parts, solving equations, expressing trigonometric functions in terms of others, verifying functions satisfy the Cauchy-Riemann equations, computing line integrals, and using contour integration techniques. The problems cover a wide range of foundational topics in complex analysis.
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Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
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Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
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A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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2. Problem 1:
Here are a few questions that you should be able to answer based only on 18.06:
a) Suppose that B is a Hermitian positive-definite matrix. Show that there is a unique
matrix √ B which is Hermitian positive-definite and has the property √( B)2 = B.
(Hint: use the diagonalization of B.)
(b) Suppose that A and B are Hermitian matrices and that B is positive-definite.
(i) Show that B−1A is similar (in the 18.06 sense) to a Hermitian matrix. (Hint: use your
answer from above.)
(ii) What does this tell you about the eigenvalues λ of B−1A , i.e. the solutions of B−1Ax =
λx?
(iii) Are the eigenvectors x orthogonal?
(iv) In Julia, make a random 5 × 5 real-symmetric matrix via A=rand(5,5); A = A+A’ and
a random 5 × 5 positive-definite matrix via B = rand(5,5); B = B’*B ... then check that
the eigenvalues of B−1A match your expectations from above via lambda,X =
eigvals(BA) (this will give an array lambda of the eigenvalues and a matrix X whose
columns are the eigenvectors).
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3. (v) Using your Julia result, what happens if you compute C = XT BX via C=X’*B*X?
You should notice that the matrix C is very special in some way. Show that the elements
Cij of C are a kind of “dot product” of the eigenvectors i and j, but with a factor of B in
the middle of the dot product.
(c) The solutions y(t) of the ODE y “− 2y ‘− cy = 0 are of the form y(t) =
for some constants C1 and C2 determined by
the initial conditions. Suppose that A is a real-symmetric 4×4 matrix with eigenvalues 3,
8, 15, 24 and corresponding eigenvectors x1, x2, . . . , x4, respectively.
(i) If x(t) solves the system of ODEs Ax with initial conditions x(0)
= a0 and x′ (0) = b0, write down the solution x(t) as a closed-form expression (no
matrix inverses or exponentials) in terms of the eigenvectors x1, x2, . . . , x4 and
a0 and b0. [Hint: expand x(t) in the basis of the eigenvectors with unknown
coefficients c1(t), . . . , c4(t), then plug into the ODE and solve for each coefficient
using the fact that the eigenvectors are _________.]
(ii) After a long time t ≫ 0, what do you expect the approximate form of the
solution to be?
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4. Problem 2:
In class, we considered the 1d Poisson equation
for the vector space of functions u(x) on x ∈ [0, L] with the “Dirichlet” boundary
conditions u(0) = u(L) = 0, and solved it in terms of the eigenfunctions of
(giving a Fourier sine series). Here, we will consider a couple of small variations on
this:
a) Suppose that we we change the boundary conditions to the periodic boundary
condition u(0) = u(L).
(i) What are the eigenfunctions of now?
(ii) Will Poisson’s equation have unique solutions? Why or why not?
(iii) Under what conditions (if any) on f(x) would a solution exist? (You can restrict
yourself to f with a convergent Fourier series.)
(b) If we instead consider
conditions v(0) = dx2 v(x) v(L) + 1, do these functions form a vector space? Why
or why not?
for functions v(x) with the boundary
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5. (c) Explain how we can transform the v(x) problem of the previous part back into the
original problem with u(0) = u(L), by writing u(x) = v(x) + q(x)
and f(x) = g(x) + r(x) for some functions q and r. (Transforming a new problem into
an old, solved one is always a useful thing to do!)
Problem 3:
For this question, you may find it helpful to refer to the notes and reading from lecture
3. Consider a finite-difference approximation of the form:
(a) Substituting the Taylor series for u(x+ Δx) etcetera (assuming u is a smooth function
with a convergent Taylor series, blah blah), show that by an appropriate choice of the
constants c and d you can make this approximation fourth-order accurate: that is, the
errors are proportional to (Δx)4 for small Δx.
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6. (b) Check your answer to the previous part by numerically computing u ′ (1) for u(x) =
sin(x), as a function of Δx, exactly as in the handout from class (refer to the notebook
posted in lecture 3 for the relevant Julia commands, and adapt them as needed). Verify
from your log-log plot of the |errors| versus Δx that you obtained the expected fourth-
order accuracy.
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7. Problem 1:
(a) Since it is Hermitian, B can be diagonalized: B = QΛQ∗, where Q is the matrix
whose columns are the eigenvectors (chosen orthonormal so that Q−1 = Q∗ ) and Λ is
the diagonal matrix of eigenvalues. Define √ Λ as the diagonal matrix of the (positive)
square roots of the eigenvalues, which is possible because the eigenvalues are > 0 (since
B is positive-definite). Then define √ B = Q √ ΛQ∗ , and by inspection we obtain (√ B)2
= B. By construction, √ B is positive-definite and Hermitian.
It is easy to see that this √ B is unique, even though the eigenvectors X are not unique,
because any acceptable transformation of Q must commute with Λ and hence with √ Λ.
Consider for simplicity the case of distinct eigenvalues: in this case, we can only scale the
eigenvectors by (nonzero) constants, corresponding to multiplying Q on the right by a
diagonal (nonsingular) matrix D. This gives the same B for any D, since QDΛ(QD)−1 =
QΛDD−1Q−1 = QΛQ−1 (diagonal matrices commute), and for the same reason it gives
the same √ B. For repeated eigenvalues λ, D can have off-diagonal elements that mix
eigenvectors of the same eigenvalue, but D still commutes with Λ because these off-
diagonal elements only appear in blocks where Λ is a multiple λI of the identity (which
commutes with anything).
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8. (b) Solutions:
(i) From 18.06, B−1A is similar to C = MB−1AM−1 for any invertible M. Let M = B1/2
from above. Then C = B−1/2AB−1/2, which is clearly Hermitian since A and B−1/2 are
Hermitian. (Why is B−1/2 Hermitian? Because B1/2 is Hermitian from above, and the
inverse of a Hermitian matrix is Hermitian.)
(ii) From 18.06, similarity means that B−1A has the same eigenvalues as C, and since C is
Hermitian these eigenvalues are real.
(iii) No, they are not (in general) orthogonal. The eigenvectors Q of C are (or can be
chosen) to be orthonormal (Q∗Q = I), but the eigenvectors of B−1A are X = M−1Q =
B−1/2Q, and hence X∗X = Q∗B−1Q = I unless B = I.
(iv) Note that there was a typo in the pset. The eigvals function returns only the
eigenvalues; you should use the eig function instead to get both eigenvalues and
eigenvectors, as explained in the Julia handout.
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9. The array lambda that you obtain in Julia should be purely real, as expected. (You might
notice that the eigenvalues are in somewhat random order, e.g. I got -8.11,3.73,1.65,
1.502,0.443. This is a side effect of how eigenvalues of non-symmetric matrices are
computed in standard linear-algebra libraries like LAPACK.) You can check
orthogonality by computing X∗X via X’*X, and the result is not a diagonal matrix (or
even close to one), hence the vectors are not orthogonal.
(v) When you compute C = X∗BX via C=X’*B*X, you should find that C is nearly
diagonal: the off-diagonal entries are all very close to zero (around 10−15 or less). They
would be exactly zero except for roundoff errors (as mentioned in class, computers keep
only around 15 significant digits). From the definition of matrix multiplication, the entry
Cij is given by the i-th row of X∗ multiplied by B, multiplied by the j-th column of X. ∗
But the j-th column X is the j-th eigenvector xj , and the i-th row of X∗ is x . Hence i ∗
Cij = xi Bxj, which looks like a dot product but with B in the middle. The fact that C
is diagonal means that which is a kind of orthogonality relation.
In fact, if we define the inner product (x, y) = x∗By, this is a perfectly good inner product
(it satisifies all the inner-product criteria because B is positive-definite), and we will see
in the next pset that B−1A is actually self-adjoint under this inner product. Hence it is no
surprise that we get real eigenvalues and orthogonal eigenvectors with respect to this
inner product.]
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10. (c) Solutions:
(i) If we write x(t) then plugging it into the ODE and using the
eigenvalue equation yields
Using the fact that the xn are necessarily orthogonal (they are eigenvectors of a
Hermitian matrix for distinct eigenvalues), we can take the dot product of both sides with
xm to find that c¨n − 2˙cn − λnc = 0 for each n, and hence
Again using orthogonality to pull out the n-th term, we find
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11. (note that we were not given that xn were normalized to unit length, and this is not
automatic) and hence we can solve for αn and βn to obtain:
(ii) After a long time, this expression will be dominated by the fastest growing term,
which √ (1+ 1+λn)t is the e term for λ4 = 24, hence:
Problem 2:
(a) Suppose that we we change the boundary conditions to the periodic boundary
condition u(0) = u(L).
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12. As in class, the eigenfunctions are sines, cosines, and exponentials, and it only remains to
apply the boundary conditions. sin(kx) is periodic if k = for n = 1, 2, . . . (excluding
n = 0 because we do not allow zero eigenfunctions and excluding n < 0 because they are
not linearly independent), and cos(kx) is periodic if n = 0, 1, 2, . . . (excluding n < 0 since
they are the same functions). The eigenvalues are −k2 = −(2πn/L)2.
is periodic only for imaginary k = but in this case we obtain
of the sin and cos eigenfunctions above. Recall from 18.06 that the eigenvectors for a
given eigenvalue form a vector space (the null space of A − λI), and when asked for
eigenvectors we only want a basis of this vector space. Alternatively, it is acceptable to
start with exponentials and call our eigenfunctions
cos(2πnx/L) + isin(2πnx/L), which is not linearly independent
which case we wouldn’t give sin and cos eigenfunctions separately.
for all integers n, in
Similarly, sin(φ + 2πnx/L) is periodic for any φ, but this is not linearly independent since
sin(φ + 2πnx/L) = sin φ cos(2πnx/L) + cos φ sin(2πnx/L).
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13. [Several of you were tempted to also allow sin(mπx/L) for odd m (not just the even m
considered above). At first glance, this seems like it satisfies the PDE and also has u(0) =
u(L) (= 0). Consider, for example, m = 1, i.e. sin(πx/L) solutions. This can’t be right,
however; e.g. it is not orthogonal to 1 = cos(0x), as required for self-adjoint problems.
The basic problem here is that if you consider the periodic extension of sin(πx/L), then it
doesn’t actually satisfy the PDE, because it has a slope discontinuity at the endpoints.
Another way of thinking about it is that periodic boundary conditions arise because we
have a PDE defined on a torus, e.g. diffusion around a circular tube, and in this case the
choice of endpoints is not unique—we can easily redefine our endpoints so that x = 0 is in
the “middle” of the domain, making it clearer that we can’t have a kink there. (This is one
of those cases where to be completely rigorous we would need to be a bit more careful
about defining the domain of our operator.)]
(ii) No, any solution will not be unique, because we now have a nonzero nullspace
spanned by the constant function u(x) = 1 (which is periodic):
Equivalently, we have a 0 eigenvalue corresponding to cos(2πnx/L) for n = 0 above.
(iii) As suggested, let us restrict ourselves to f(x) with a convergent Fourier series. That
is, as in class, we are expanding f(x) in terms of the eigenfunctions:
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14. (You could also write out the Fourier series in terms of sines and cosines, but the
complexexponential form is more compact so I will use it here.) Here, the coefficients cn,
by the usual orthogonality properties of the Fourier series, or equivalently by self-
adjointness of
In order to solve as in class we would divide each term by its eigenvalue
−(2πn/L)2, but we can only do this for n 0. Hence, we can only solve the equation if
the n = 0 term is absent, i.e. c0 = 0. Appling the explicit formula for c0, the equation is
solvable (for f with a Fourier series) if and only if:
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15. There are other ways to come to the same conclusion. For example, we could expand
u(x) in a Fourier series (i.e. in the eigenfunction basis), apply d2/dx2, and ask what is
the column space of d2/dx2? Again, we would find that upon taking the second
derivative the n = 0 (constant) term vanishes, and so the column space consist of Fourier
series missing a constant term.
The same reasoning works if you write out the Fourier series in terms of sin and cos
sums separately, in which case you find that f must be missing the n = 0 cosine term,
giving the same result.
(b) No. For example, the function 0 (which must be in any vector space) does not
satisy those boundary conditions. (Also adding functions doesn’t work, scaling them
by constants, etcetera.)
(c) We merely pick any twice-differentiable function q(x) with q(L) − q(0) = −1, in
which case u(L) − u(0) = [v(L) − v(0)] + [q(L) − q(0)] = 1 − 1 = 0 and u is periodic.
Then, plugging v = u − q into we obtain
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16. which is the (periodic-u) Poisson equation for u with a (possibly) modified right-hand
side.
For example, the simplest such q is probably q(x) = x/L, in which case d2q/dx2 = 0 and
u solves the Poisson equation with an unmodified right-hand side.
Problem 3:
We are using a difference approximation of the form:
(a) First, we Taylor expand:
The numerator of the difference formula flips sign if Δx → −Δx, which means that when
you plug in the Taylor series all of the even powers of Δx must cancel! To get 4th-order
accuracy, the Δx3 term in the numerator (which would give an error ∼ Δx2) must cancel
as well, and this determines our choice of c: the Δx3 term in the numerator is
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17. and hence we must have c = 23 = 8
are the Δx term and the Δx5 term:
The remaining terms in the numerator
Clearly, to get the correct u' (x) as Δx → 0, we must have d = 12
Hence, the error is approximately
which is ∼ Δx4 as desired.
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18. Figure 1: Actual vs. predicted error for problem 1(b), using fourth-order difference
approximation for u' (x) with u(x) = sin(x), at x = 1.
b) The Julia code is the same as in the handout, except now we compute our difference
approximation by the command: d = (-sin(x+2*dx) + 8*sin(x+dx) - 8*sin(x+dx) +
sin(x-2*dx)) ./ (12 * dx); the result is plotted in Fig. 1. Note that the error falls as a
straight line (a power law), until it reaches ∼ 10−15, when it starts becoming
dominated by roundoff errors (and actually gets worse). To verify the order of
accuracy, it would be sufficient to check the slope of the straight-line region, but it is
more fun to plot the actual predicted error from the previous part, where
Clearly the predicted error is almost exactly
right (until roundoff errors take over).
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