The universe is a song, singing itself.

No, really. The solutions of the Schrödinger Equation are harmonics, just like musical notes.

Quantum state of an electron orbiting a hydrogen atom where n=6, l=4, and m=1 (spin doesn’t matter):

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This is an equal superposition of the |3,2,1> and |3,1,-1> eigenstates:

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This is an equal superposition of the |3,2,2> and |3,1,-1> eigenstates:

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This is an equal superposition of the |4,3,3> and |4,1,0> eigenstates:

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What is an eigenstate? It’s a convenient state to use as a basis. We get to decide which quantum states are “pure” and which “mixed”. There’s an easy way and a hard way; the easy way is to use eigenstates as the pure states.

More mathematically: the Schrödinger equation tells us what’s going on in an atom. The answers to the Schrödinger equation are complex and hard to compare. But phrasing the answers as combinations of eigenfunctions makes them comparable. One atom is 30% state A, 15% state B, 22% state C … and another atom is different percentages of the same states.

Just like vectors in 3-D space, where you can orient the axes differently – you can pick different directions for x, y, and z to point in. But now the vectors are abstract, representing states. Still addable so still vectors. Convex or linear combinations of those “pure” states describe the “mixed” states.


SOURCE: Atom in a Box

Eigenvector Example

Introduction to eigenvector threat:

An Eigenvector is defined as a non-zero vector irruptive which we can’t change its direction by use of a given linear transformation. Dead straight transformation can occur denoted as T. Successional progressivism can be given ad eundem follows, T(v)= `lambdav`.The unassociated name of Eigen vector is characteristic vector. Eigen vector produces the scalar multiplication of the original virus. Eigenvector has a wide range of applications incoming all over the fields.In this introductory study we are going to clap eyes on virtuoso examples for eigen phytogenic infection.

Computation of eigen vector embodiment:

‚¬ A linear transformation T: Rn that tends against Rn given by an n x n womb B. The Eigen value l and the eigenvector v of T hind end be defined by Bv = lv.

‚¬ Unvaryingly, v is a vector that has in purportless space (B- lI). The chiliad l and the hand infection v are also called the Eigen value and the eigenvector of B.

‚¬ The following proposes may helps to find the Eigen values.

‚¬ electric railway is an Eigen pleasantness of matrix B.

‚¬ Bv = lv where v should not endure equal to nada.

‚¬ (B-lI)x = 0.that has a non trivial solution sign manual=v.

‚¬ B-lI is non invertible.

‚¬ Determination in regard to B-lI = 0.

‚¬ The characteristic polynomial with regard to a given square matrix B is det(B-lI).

‚¬ Thus the Eigen values and the eigenvectors can be work out as follows.

Step 1: Get the Eigen values l1 and l2 by calculating the characteristics equation.

Step 2: For every Eigen denotation backstage solve the homogeneous long-range plan B-lI = 0.

and puzzle the eigenvectors with li as the Eigen value.

Example Problems for Eigenvector:

Eigen vector itemize 1:

If that B is a matrix and that inverse matrix of B is B^-1 and if that y is an eigenvector for die B with the Eigen value is `]]2,1],]4,4]]` €° 0. Prove that y is an eigenvector in lieu of inverse matrix B^-1 with the Eigen value `]]2,1],]4,4]]`^-1 (inverse of matrix `]]2,1],]4,4]]`).


Let us gain B.y = c, therefore: y = B^-1 c

Where B is a impression
When a prototype B and a nonzero contagiousness y satisfy: B.y = `]]2,1],]4,4]]` y (for some scalar matrix `]]2,1],]4,4]]`), and therefore y = B^-1 c,

Then we journey the denotation of y as follows,

y= `]]2,1],]4,4]]`-1.y,

wherefore: `]]2,1],]4,4]]`^-1.y = B^-1.y

Eigen vector example 2:

Consider the following 2x2 seal


Revelation all the eigenvectors that are patrilateral to the Eigen quality `lambda=3`


Open arms the above authenticated case we have verified that in actuality `lambda=3` is an Eigen atmosphere touching the given matrix. Draw off Y0 be an eigenvector that are parallel to the Eigen value `lambda=3`.

Set Y0= `]]x0,],]yo,]]`. Then we have the following equations

(2-3)x0 + -y0 = 0.

0 + (3-3)y0 = 0.

which reduces to the companionless denominator

-x0-y0 = 0.

This yields y = -x. Therefore, we have

Y0= `]]x0,],]yo,]]` = `]]x0,],]-xo,]]`

Y0=x0 `]]1,],]-1,]]`

Remain that we are all having all in point of the eigenvectors that are of that ilk to the Eigen value `lambda=3`.

transformations on transformations on transformations

part 5 of linear algebra (toc)

stacks on stacks on stacks. racks on racks on racks. cats in hats on knox in box. Hey classy people, we’re rocking diagonalisation and markov chains today! (That means numbers. great.) Inherently included is matrix representation and the change of basis matrix mindboggle… I’ll briefly touch on those.

So we’ll be a bit numerical this time around. Let’s talk matrices. (I want to get through this stuff quickly so…speed time aiight guys?) How do we represent a transformation in Euclidean space with a matrix? It’s kind of an interesting question. For one, how do we know we can? For two, how do we know the matrix representation is unique?

Keep reading

Singular Value Decomposition


[Click here for a PDF of this post with nicer formatting]

We’ve been presented with the definition of the singular value decomposition or SVD, but not how to compute it.

Recall that the definition was

Singular value decomposition (SVD)

Given \( M \in \text{R}^{m \times n} \), we can find a representation of \( M \)

M = U \Sigma V^\T,

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plus ça change, plus c'est la même chose

part 4 on linear algebra (toc)

It’s been a while. whoops. So I’m going to do eigenvalues and eigenvectors today. These babies are an example of something that legitimately gets applied quite a lot outside of the realm of math, in (system? controls?) engineering. But…hey, we’re all math majors here. Who cares about practical application?

(Side note for anyone who cares: I will be flying back to the rainy west tomorrow, and school starts in a week for me. We’ll be hurling into complex analysis and abstract algebra territories soon. Can you barely contain your excitement? Because I can’t. It’s exploding out of me like acid reflux.)

Keep reading

What do eigenvalues and eigenvectors represent intuitively?
Answer by Karolis Juodele:

Eigenvectors and eigenvalues are not something you use in a matrix. It’s something a matrix has.

Eigenvector is a vector which, multiplied by the matrix does not change direction. It may change length. Eigenvalue of an eigenvector is the magnitude by which the vector length changed.

One example, the rotation matrix rotates all vectors except those that match its rotation axis. These vectors are its eigenvectors and their eigenvalues are 1. Another example could be a scaling matrix which doubles the length of any vector. All vectors are its eigenvectors and their eigenvalues are equal to 2.

What do eigenvalues and eigenvectors represent intuitively?

Introduction to Linear Dynamical Systems
Stephen Boyd
Genre: Advanced Mathematics
Price: Get
Publish Date: July 9, 2008

Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems. Topics include: Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Matrix exponential, stability, and asymptotic behavior. Multi-input multi-output systems, impulse and step matrices; convolution and transfer matrix descriptions. Control, reachability, state transfer, and least-norm inputs. Observability and least-squares state estimation. http://dlvr.it/KSkYsq

[1602.02896] Anderson localisation for infinitely many interacting particles in Hartree-Fock theory

[ Authors ]
Raphael Ducatez
[ Abstract ]
We prove the occurrence of Anderson localisation for a system of infinitely many particles interacting with a short range potential, within the ground state Hartree-Fock approximation. We assume that the particles hop on a discrete lattice and that they are submitted to an external periodic potential which creates a gap in the non-interacting one particle Hamiltonian. We also assume that the interaction is weak enough to preserve a gap. We prove that the mean-field operator has exponentially localised eigenvectors, either on its whole spectrum or at the edges of its bands, depending on the strength of the disorder.

MTH309 Introduction to Linear Algebra
Bernard Badzioch
Genre: Advanced Mathematics
Price: Get
Publish Date: June 27, 2011

These are materials for the course MTH 309 Introduction to Linear Algebra. Topics covered by this course include: systems of linear equations; matrix algebra; determinants of matrices; vector spaces and linear transformations; eigenvalues and eigenvectors; inner product spaces.

This work is licensed under the Creative Commons Attribution-NonCommercial-Share

[1602.00909] Rydberg systems in parallel electric and magnetic fields: an improved method for finding exceptional points

[ Authors ]
Matthias Feldmaier, Jörg Main, Frank Schweiner, Holger Cartarius, Günter Wunner
[ Abstract ]
Exceptional points are special parameter points in spectra of open quantum systems, at which resonance energies degenerate and the associated eigenvectors coalesce. Typical examples are Rydberg systems in parallel electric and magnetic fields, for which we solve the Schr"odinger equation in a complete basis to calculate the resonances and eigenvectors. Starting from an avoided crossing within the parameter-dependent spectra and using a two-dimensional matrix model, we develop an iterative algorithm to calculate the field strengths and resonance energies of exceptional points and to verify their basic properties. Additionally, we are able to visualise the wave functions of the degenerate states. We report the existence of various exceptional points. For the hydrogen atom these points are in an experimentally inaccessible regime of field strengths. However, excitons in cuprous oxide in parallel electric and magnetic fields, i. e., the corresponding hydrogen analogue in a solid state body, provide a suitable system, where the high-field regime can be reached at much smaller external fields and for which we propose an experiment to detect exceptional points.

[1602.00718] Low-frequency electromagnetic field in a Wigner crystal

[ Authors ]
Anton Stupka
[ Abstract ]
Long-wave low-frequency oscillations are described in a Wigner crystal by generalization of the reverse continuum model for the case of electronic lattice. The internal self-consistent long-wave electromagnetic field is used to describe the collective motions in the system. The eigenvectors and eigenvalues of the obtained system of equations are derived. The velocities of longitudinal and transversal sound waves are found.

What is the importance of determinants in linear algebra?
Answer by Sam Lichtenstein:

Short version: Yes, determinants are useful and important. No, they are not necessary for defining the basic notions of linear algebra, such linear independence and basis and eigenvector, or the concept of an invertible linear transformation (or matrix). They are also not necessary for proving most properties of these notions. But yes, I think a good course on linear algebra should still introduce them early on.

Long version: Determinants are a misunderstood beast. It’s only natural: they are computed via an extremely ugly (to my eye) formula, or a recursive algorithm (expansion by minors), both of which involve annoying signs that can be difficult to remember. But as Disney taught us, a beast can have a heart of gold and talking cutlery.

First, though, I emphasize that the determinant is not strictly necessary to get started in linear algebra. For a thorough explanation of this, see Axler’s article Down with Determinants (http://www.axler.net/DwD.pdf), and his textbook Linear algebra done right. This explains the pedagogical decision by some authors to postpone treating determinants until later chapters of their texts: the complicated formula and the mechanics of working with determinants are simply a distraction from one’s initial goals in linear algebra (learning about vectors, linear transformations, bases, etc).

Yes the later chapters are still crucial.

Fundamentally, determinants are about volume. That is, they generalize and improve the notion of the volume of the parallelipiped (= higher dimension version of a parallelogram) swept out by a collection of vectors in space.  This is not the place to give a treatise on exterior algebra, the modern language via which mathematicians explain this property of determinants, so I refer you to the eponymous Wikipedia article. The subtle point is that while we are used to thinking of vector spaces as n-dimensional Euclidean space (R^n), with volume defined in terms of the usual notion of distance (the standard inner product on R^n), in fact vector spaces are a more abstract and general notion. They can be endowed with alternate notions of distance (other inner products), and can even be defined over fields other than the real numbers (such as the rational numbers, the complex numbers, or a finite field Z/p). In such contexts, volume can still be defined, but not “canonically”: you have to make a choice (of an element in the 1-dimensional top exterior power of your vector space). You can think of this as fixing a scale. The useful property of determinants is that while the scale you fix is arbitrary, the volume-changing effect of a linear transformation of your vector space is independent of the choice of scale: it is exactly the determinant of said linear transformation. This is why the answer to your question, “Are there any real-life applications of determinants?” is undoubtedly yes. They arise all the time as normalization factors, because it is often a good idea to preserve the scale as you perform operations on vectors (such as data points in R^n). [This can be important, for example, to preserve the efficacy or improve the efficiency of numerical algorithms.]

Now what about the applications you mention, such as testing linear independence of a set of n vectors in an n-dimensional vector space (check if the determinant is nonzero), or inverting a matrix (via Cramer’s rule, which involves a determinant), or – to add another – finding eigenvalues of a matrix (roots of a characteristic polynomial, computed as a determinant)?  These are all reasonable things to do, but in practice I believe they are not very efficient methods for accomplishing the stated goals. They become slow and unwieldy for large matrices, for example, and other algorithms are preferred. Nonetheless, I firmly believe that everyone should know how to perform these tasks, and get comfortable doing them by hand for 2 by 2 and 3 by 3 matrices, if only to better understand the concepts involved. If you cannot work out the eigenvalues of a 2 by 2 matrix by hand, then you probably don’t understand the concept, and for a “general” 2 by 2 matrix a good way to do it quickly is to compute the characteristic polynomial using the “ad-bc” formula for a 2 by 2 determinant.

You ask whether determinants have other uses in linear algebra. Of course they do. I would say, in fact, that they are ubiquitous in linear algebra. This ubiquity makes it hard for me to pin down specific examples, or to point to nice motivating examples for your students. But here is a high-brow application in abstract mathematics. Given a polynomial [math]a_n x^n + \cdots + a_1 x + a_0[/math], how can we tell if it has repeated roots without actually factoring it or otherwise finding the roots? In fact, there is an invariant called the discriminant which gives the answer. A certain polynomial function [math]\Delta(a_0,\ldots, a_n)[/math] can be computed, and this vanishes if and only if the original polynomial has a repeated root.  Where does the discriminant come from? It is essentially the determinant of a rather complicated matrix cooked up from the numbers [math]a_0,\ldots, a_n[/math].

A more down-to-earth application that might be motivating for some students is the Jacobian determinant that enters, for example, into change-of-variables formulas when studying integrals in multivariable calculus. If you ever find yourself needing to to work with spherical coordinates and wonder why an integral with respect to [math]d x d y d z[/math] becomes an integral with respect to [math]\rho^2 \sin \phi d\rho d\phi d\theta[/math], the answer is: a certain determinant is equal to [math]\rho^2 \sin \phi[/math]. Of course, depending on the university, a course in linear algebra might precede multivariable calculus, which would make this “motivating example” less useful.

Another remark to make is that for many theoretical purposes, it suffices simply to know that there is a “nice” formula for the determinant of a matrix (namely, a certain complicated polynomial function of the matrix entries), but the precise formula itself is irrelevant. For example, many mathematicians use constantly the fact that the set of polynomials of a given degree which have repeated roots, is “cut out” from the set of all polynomials of that degree, by a condition on the coefficients which is itself polynomial; indeed, this is the discriminant I mentioned above. But they rarely care about the formula for the discriminant, merely using the fact that one exists. E.g., simply knowing there is such a formula tells you that polynomials with repeated roots are very rare, and in some sense pathological, but that if you are unlucky enough to get such a guy, you can get a nice polynomial with distinct roots simply by wiggling all the coefficients a bit (adding 0.00001 to any coefficient will usually work). [I can expand on this if you post another question on the subject.]

What is the importance of determinants in linear algebra?
When big data gets too big, this machine-learning algorithm may be the answer
Big data may hold a world of untapped potential, but what happens when your data set is bigger than your processing power can handle? A new algorithm that taps quantum computing may be able to help.
By Katherine Noyes

A new quantum computing algorithm combines topology with quantum computing. Never mind that it probably requires more qubits than any currently existing quantum computer has. But who knows, because in the paper they never say how many qubits you need – they always just say n qubits. Specifically, what they did was invent a quantum computing algorithm to calculate Betti numbers, which count the numbers of connected components, holes, and voids, in a geometric structure, along with the eigenvectors and eigenvalues of the combinatorial Laplacian. The combinatorial Laplacian matrix is a matrix representation of a graph, the eigenvector is the characteristic vector that does not change its direction when transformed by the matrix, and the eigenvalue is the scalar that goes with the eigenvector to make the the same result as the eigenvector transformed by the matrix. The algorithm is related to quantum matrix inversion algorithms.

“Diagonalizing a 2^n by 2^n sparse matrix using a quantum computer takes time O(n^2), compared with time O(2^2n) on a classical computer.”

Moving on to more general non-convex optimization, in my talk, I pointed out the difficulty in even converging to a local optimum due to the existence of saddle points. Saddle points are critical points which are not local minima, meaning there exist directions where the objective value decreases (for minimization problems). Saddle points can slow down gradient descent arbitrarily. Alternatively, if Newton’s method is run, it converges to an arbitrary critical point, and does not distinguish between a local minimum and a saddle point.

One solution to escape saddle points is to use the second order Hessian information to find the direction of escape when the gradient value is small: the Hessian eigenvectors with negative eigenvalues provide such directions of escape. See works here, here and here. A recent work surprisingly shows that it is possible to escape saddle points using only first order information based on noisy stochastic gradient descent (SGD). In many applications, this is far cheaper than (approximate) computation of the Hessian eigenvectors. However, one unresolved issue is handling degenerate saddle points, where there are only positive and zero eigenvalues in the Hessian matrix. For such points, even distinguishing saddle points from local optima is hard. It is also an open problem to establish the presence or absence of such degenerate saddle points for particular non-convex problems, e.g. in deep learning.

Differential Equations, Spring 2006 - Instructors: Prof. Arthur Mattuck Prof. …:

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Differential Equations, Spring 2006
Instructors: Prof. Arthur Mattuck Prof. Haynes Miller
Genre: Mathematics
Price: Get
Publish Date: June 27, 2007

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODEs) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODEs by analytical, graphical and numerical methods; Linear ODEs, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams.

http://ocw.mit.edu; Creative Commons Attribution-NonCommercial-ShareAlike 2.5; h