User jiahao chen - MathOverflow

Why were matrix determinants once such a big deal?

2010-08-18T17:09:48Z

I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. I am curious about why determinants were such a big deal. So here's my question:

a) What are examples of cool tricks you can use matrix determinants for? (Cramer's rule comes to mind, but I can't come up with much more.) What kind of amazing properties do matrix determinants have that made them such popular objects of study?

b) Why did the use of matrix determinants fall out of favor? Some back history would be very welcome.

Update: From the responses below, it seems appropriate to turn this question into a community wiki. I think it would be useful to generalize the original series of questions with:

c) What significance do matrix determinants have for other branches of mathematics? (For example, the geometric significance of the determinant as the signed volume of a parallelepiped.) What developments in mathematics have been inspired by/aided by the theory of matrix determinants?

d) For computational and theoretical applications that matrix determinants are no longer widely used for today, what have supplanted them?

Is there a name for the matrix equation A X B + B X A + C X C = D?

2009-11-19T20:29:41Z

I happen to be working on a problem that reduces to solving the following equation:

$$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$

where A through D are known matrices ( A, B, D are real, symmetric matrices and C is real and antisymmetric), and X is an unknown square matrix to be solved for.

Is there a name for this equation, and is there any known algorithm for solving this equation? (Without the C X C term this reduces to the continuous Lyapunov equation given either A or B is an invertible matrix. I wonder if anyone working in control theory may have seen such equations before.)

Random Schrödinger operators with asymmetric Lifshitz tails?

2012-09-13T03:23:42Z

For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ that are less than or equal to $E$. (Or see the rigorous definition in Sec. 2 of Ref. 1.)

The Lifshitz exponent $\gamma_-$ is defined near the infimum $E_- = \inf \sigma(H)$ as (e.g. in Ref. 1):

$$\gamma_- = \lim_{E\downarrow E_-} \frac {\ln (-\ln N(E))}{\ln(E-E_-)},$$

and similarly near the supremum $E_+ = \sup \sigma(H)$, the analogous exponent can be defined as

$$\gamma_+ = \lim_{E\uparrow E_+} \frac {\ln (-\ln (1-N(E)))}{\ln(E_+-E)}.$$

These exponents measure how fast the density of eigenvalues grows or decays near the extreme ends of the spectrum.

I have some empirical data for a disordered quantum mechanical system for which I appear to measure $\gamma_- \ne \gamma_+$. This behavior does not seem to be observed in the model Hamiltonians that have been summarized in the review in Ref. 1 or in its references. I haven't found any literature that describes such behavior in any theoretical studies of quantum mechanical systems or random Schrödinger operators.

As I am not an expert in this field, I would appreciate any pointers to references where such asymmetric behavior in the upper and lower tails of $N(E)$ have been observed.

Reference

Kirsch and Metzger, http://arxiv.org/abs/math-ph/0608066

Answer by Jiahao Chen for Expected values of traces of products of random matrices

2012-09-06T04:25:41Z

For the unitary group, the first paper I am aware of to do these sorts of averages is:

http://link.aip.org/link/JMAPAQ/v21/i12/p2695/s1

An early paper of Collins' in 2003 expresses such averages in terms of Weingarten functions, which are usually expressed as character expansions over $U(N)$, $O(N)$ or $Sp(N)$.

http://arxiv.org/abs/math-ph/0205010

Some early papers calculating these character expansions are:

http://jmp.aip.org/resource/1/jmapaq/v25/i6/p2028_s1

http://jmp.aip.org/resource/1/jmapaq/v43/i1/p604_s1

Reconstructing an Euclidean point cloud from their pairwise distances

2012-05-21T23:50:23Z

I have a collection of points $P_1, ..., P_N$ in some Euclidean space $\mathbb R^m$ and the coordinates $x_1, x_2, ..., x_N$ respectively associated with them, where $x_i$ is the usual Cartesian tuple $x_i = (x_{i1}, x_{i2}, ..., x_{im})$. I can immediately calculate the pairwise distances between these points $r_{12}, r_{13}, ..., r_{N-1,N}$ under the usual Euclidean norm using Pythagoras' theorem (in $m$ dimensions), i.e. $r_{ij} = \left\Vert x_i - x_j \right\Vert$.

Suppose now I have the converse situation where I have the points $P_1, ..., P_N$ and all their associated pairwise distances $\{r_{ij}\}$, and I don't know their coordinates. What is known about the embeddability of these points in an Euclidean space, and is it possible to reconstruct the Cartesian coordinates for them? In other words, given an arbitrary collection of nonnegative numbers $\{r_{ij}\}$, how do I find all positive integers $m$ and enumerate all the possible sets of $N$ coordinates $x_1, x_2, ..., x_N \in \mathbb R^m$ that are consistent with the interpretation $r_{ij} = \left\Vert x_i - x_j \right\Vert$?

How to quantify noncommutativity?

2012-05-11T17:39:14Z

If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for finite complex-valued matrices $A, B \in \mathbb C^{n\times n}$.)

Let me try the obvious thing here: by definition if $A$ and $B$ commute, then the commutator $[A, B] = AB-BA = 0$. Naively would use some sort of functional like an operator norm to reduce this to a number that could potentially behave like a metric. The first thing I thought of was the trace, but clearly that doesn't work since $\mathrm{tr } [A, B] =\mathrm{tr } (AB-BA) = \mathrm{tr }AB - \mathrm{tr }AB = 0$ always. One could then turn to, say, the Frobenius norm of $[A, B]$. What is known about the maximal (or supremal) value of such norms?

Are there quantifiers of noncommutativity that can also account for higher-order effects, e.g. cases where $[A, B] \ne 0$ but $[A, [A,B]] = 0$? This should be "less" non-commuting than if $[A, B] \ne 0$ and $[A, [A,B]] \ne 0$ and $[B, [A,B]] \ne 0$ but, say, $[A, [B, [A, B]]] = 0$.

For those who prefer a free algebraic setting, the question can be framed as: how free is a non-free algebra? Is there a sensible way to measure proximity to a free algebra? What if I had an algebra where $AB=BA$ is the only one relation that makes it not a free algebra; is there a sense it is "less free" or "more free" than an algebra where $ABABAB=BAA$ is the only such relation, or example.

Motivation: it is sometimes said that free probability is the study of "maximally" non-commuting objects. I would like to know if this statement can be made precise in the sense of how one can define "maximally non-commuting" in a sensible fashion.

Combinatorial interpretation of the power of a series

2012-05-12T20:59:31Z

I am trying to understand a result involving the power of a series that occurs in Gradstein and Ryzhik's Table of Integrals, Series, and Products. Result 0.314 (p.17, 7th ed.) is:

$$\left(\sum_{k=0}^\infty a_k x^k\right)^n=\sum_{k=0}^\infty c_k x^k$$

where $$c_0 = a_0^n, c_m=\frac 1 {ma_0} \sum_{k=1}^m (kn-m+k) a_k c_{m-k}$$ for $m\ge1$ and $n\in\mathbb N$.

What is an appropriate combinatorial interpretation of this result?

One way I am trying to understand it is to see how it arises from the multinomial expansion

$$ \left(\sum_{k=0}^\infty b_k \right)^n = \sum_{\kappa\vdash k} \binom k \kappa b^\kappa $$

which has the usual nice combinatorial interpretation of how to put objects in bins. This is suggested in the reference given in Gradstein and Ryzhik, which is an even older book: Smithsonian mathematical formulae and tables of elliptic functions, p.118. However, the additional structure provided by regrouping powers of $x$ after substituting $b_k = a_k x^k$ must surely have some significant, nontrivial and well-known combinatorial implications that I am simply unaware of. (I hope this is clear; the multiindex notation is new to me and I don't know a nice way to write the result of this last step.)

Generalizations of Gram-Charlier and Edgeworth series?

2012-02-09T22:22:29Z

I am looking for references pertaining to, and/or help in deriving, generalizations of the Gram-Charlier and Edgeworth series for non-Gaussian reference probability distributions.

I would like to outline here a sketch of an approach I am taking in deriving a generalization, which could help explain the problems I am having. I am hoping that this might remind someone of something useful which I should be looking into for this problem.

I am considering a generalization of the exposition in, e.g. Kendall's Advanced Theory of Statistics, vol. 1, Section 6.20 and thereabouts. Consider two formal series expansions of a probability density function $f(x)$ in orthogonal polynomials of the forms

$$ f(x) = \sum_{n=0}^\infty c_n \phi_n(x) w(x) $$ $$ f(x) = g\left(\sum_{n=0}^\infty k_n D_n\right) w(x) $$

where $\phi_n$ are the orthogonal polynomials related to the weight function $w$ and by construction we have $c_n$ is the expectation of the $n$th orthogonal polynomial, i.e. $$ c_n = \mathbb E_f [\phi_n] = \int_{\mathbb R} \phi_n (x) f(x) dx$$

and in the second equation $D_n$ is some differential operator.

The usual Gram-Charlier or Edgeworth series is recovered when $w$ is the standard Gaussian, $g(z) = e^z$, $D_n = (-d/dx)^n $ and $\phi_n$ are the Hermite polynomials with suitable normalization. Then the coefficients $k_n$ are just

$$k_n = \frac{\kappa_n}{n!} $$

where $\kappa_n$ are the cumulants of the probability distribution $f$.

My first attempt at a generalization is to ask for what suitable $g(z)$ and $D$ to use given $w$, and what these coefficients $k_n$ become. There are standard techniques for construction the orthogonal polynomial family $\phi_n$ for a given $w$ so this determines the coefficients $c_n$. But keeping $g$ and $D$ unchanged from the Gaussian case seems to cause problems for distributions with compact support.

Consider the semicircle as a concrete example. The corresponding orthogonal polynomials are Chebyshev polynomials of the second kind $U_n$. However if we expand the second equation to just the first two terms,

$$ f(x) = \left(k_0 - k_1 \frac d {dx}\right) w(x)$$

the leading term involves $w^\prime$ which blows up at the edges of the semicircle. This does not seem to produce a useful series in the sense of being able to describe small corrections to $w$. (I realize that G-C and Edgeworth themselves don't always converge nicely either; I am ignoring this for now.) However, the Rodrigues formula for the Chebyshev polynomials appear to suggest that $D_n$ = (some constant depending on $n$) $\times \frac{d^n}{dx^n} (1-x^2)^n$ is more useful, as for the semicircle $w(x) \propto \sqrt{1-x^2}$,

$$D_n w(x) = U_n(x) w(x) $$

However, I got stuck trying to figure out $g(z)$ and $k_n$ should be.

Intuition for Haar measure of random matrix

2011-09-24T22:33:16Z

What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices?

My understanding for what Haar measure means for $U(1)$ is that it can be thought of as a measure over a uniform distribution of phases on a circle, i.e. a matrix representing $M \in U(1)$ can be parameterized with an angle $\theta$ so that $$d\mu(M) = \frac {d\theta}{2\pi} $$

What is a correct generalization of this intuition to $N>1$? In particular, are there any explicit parameterizations of Haar measure that resemble writing down angles that are uniformly distributed?

One possibility that came to mind is that eigenvectors, ~~rows and/or columns~~ have (generalized) phases that can be thought of as direction angles that are in some sense uniform over $SO(N)$ or $U(N)$? (Edit: seems like this would not be the case for rows and columns.)

Another possibility I have thought of is using Givens rotations to parameterize an orthogonal matrix using the resulting rotation angles that bring it to the identity matrix. Are there any known results about the distribution of the Givens angles? It would seem plausible that they could be uniformly distributed, but given that the Givens rotations are usually applied in a particular fashion to achieve diagonalization, that could introduce correlations that would result in nonuniformity.

(Caveat: I'm new to all this so I could very well be wrong about even trying to conceptualize such a question.)

Relationship between free probability and deterministic graphs?

2011-09-25T17:45:43Z

Consider the $N\times N$ matrix $$ M = \left(\begin{array} \\ 0 & 1 & & 0 \\ 1 & \ddots & \ddots & \\ & \ddots & \ddots & 1 \\ 0 & & 1 & 0 \\ \end{array}\right) $$

which comes from the adjacency matrix of a graph corresponding to a one-dimensional chain of $N$ nodes with dangling ends. A cartoon of this graph is $$\circ -\circ -\circ -\circ -\cdots-\circ -\circ$$

It turns out that if you plot a histogram of its eigenvalues, it appears to fit exactly with an arcsine distribution $$f(x) = \frac{1} {\pi \sqrt{4-x^2}}, \vert x \vert < 2 $$ which is exactly what one would expect from the free convolution of the binomial distribution $$ p(x) = \frac 1 2 \left( \delta\left(x-1\right) + \delta \left(x+1\right)\right)$$ with itself.

Is this mere coincidence, or evidence of something deeper? I feel like this must be some example of a known result out there.

I've gotten as far as figuring out how $\pm 1$ shows up; you can write $M$ as the sum of two pieces $$ M = A + B $$ $$ A = \left(\begin{array}{cccccc} 0 & 1\\ 1 & 0\\ & & 0 & 1\\ & & 1 & 0\\ & & & & \ddots\\ & & & & & \ddots \end{array}\right) = \sigma_x \oplus \sigma_x \oplus \cdots $$ $$ B = \left(\begin{array}{cccccc} 0\\ & 0 & 1\\ & 1 & 0\\ & & & 0 & 1\\ & & & 1 & 0\\ & & & & & \ddots \end{array}\right) = [0] \oplus \sigma_x \oplus \sigma_x \oplus \cdots $$

where $\sigma_x$ is the Pauli sigma matrix which of course has eigenvalues $\pm 1$. It must be that these two matrices are freely independent in the $N\rightarrow \infty$ limit, and possibly even for finite $N$ also, so that this reduces to the free convolution described above.

I may be reading too much into this, but it's interesting to me that this is a completely deterministic matrix problem with free probabilistic characteristics. I'm not at all familiar with the algebraic aspects of free probability theory, let alone what the graph theoretic relationships would be.

Statistics for Haar measure of random matrices?

2011-10-06T05:37:23Z

Let's say I have $M$ samples of $N\times N$ real orthogonal matrices. What statistics can I calculate to test if they could have been drawn from a distribution consistent with Haar measure over $O(N)$?

This question is related to this previous question.

Relationship between R-transform and free convolution of random matrices?

2011-09-24T19:59:19Z

I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to noncrossing partitions.

As far as I understand it, the R-transform consists of the following steps:

Given a probability distribution function $f(t)$ over some domain $D$ (which I usually take to be $\mathbb R$), find its Cauchy transform $$ g(s) = \int_D \frac {f(t)} {t-s} dt $$
Calculate the functional inverse $ g^{-1}(w) $ and subtract $\frac 1 w$ to obtain the R-transform $$ r(w) = g^{-1}(w) - \frac 1 w $$

and that the free convolution of two pdfs $f_1 \boxplus f_2$ consists of:

Adding the two R-transformed functions $$ r_s (w) = r_1(w) + r_2(w) $$
Adding $\frac 1 w$ to the sum, then computing the functional inverse of $$ g_s^{-1}(w) = r_s(w)+\frac 1 w$$
Computing the inverse Cauchy transform using the Plemelj relation $$ f_s(t) = \frac 1 \pi \Im g_s (s) $$

I am trying to understand the mechanics of R-transform (why does it work?) and relating it to calculating the free convolution using random matrices, i.e. that if we have matrices $A$ and $B$ with eigenvalue spectra $f_A$ and $f_B$, that by taking a random orthogonal matrix $Q$ of Haar measure you can form the sum $$A + Q B Q^T$$ that has spectrum $f_A \boxplus f_B $ in the large $N$ limit.

I can see that expanding the resolvent in the Cauchy transform produces a formal power series in $s$, but I don't really see how the coefficients come out to free cumulants. What is $\frac 1 w$ doing? What is the relationship with noncrossing partitions? My intuition is that they must play a role in counting the number of terms with the same value in this expansion, and that the noncrossing must arise from where the $Q$ and $Q^T$s show up in the series (effecting a change of basis), but I don't quite see it yet.

Why only three classical matrix ensembles in RMT? (Newbie question)

2010-12-29T22:09:11Z

I am just starting out on understanding random matrix theory from a background in applied mathematics. I have a very basic question about the Gaussian ensembles: why are there only three classical Gaussian ensembles? This seems very mysterious to me. Is it historically motivated from applications, or is there a deeper reason? I thought it might arise from exhausting all possible classes of diagonalizable matrices of a certain symmetry, but I have no idea if this is true or not.

I haven't been able to find a good expository reference for this, so any thoughts along those lines are also welcome.

Which graphs have incidence matrices of full rank?

2009-11-14T19:04:14Z

This is a follow-up to a previous question. What graphs have incidence matrices of full rank?

Obvious members of the class: complete graphs.

Obvious counterexamples: Graph with more than two vertices but only one edge.

I'm tempted to guess that the answer is graphs that contain spanning trees as subgraphs. However, I haven't put much thought into this.

Answer by Jiahao Chen for Applied linear algebra textbook?

2011-01-02T01:49:53Z

"Numerical Linear Algebra" by Trefethen and Bau is IMO the single best book to start learning from. It is lucidly written, concise and relatively inexpensive. Perhaps its main drawback is an unconventional presentation starting from singular value decomposition (SVD) and presenting the other standard transformations as derivatives of SVD. It worked for me though.

There are many other excellent books out there, but any good book should cover the basics like Gaussian elimination, Cholesky factorization, LU and QR decompositions, Householder reflections and Givens rotations as an absolutely bare minimum. Also essential are applications to solving linear systems, least squares problems and eigenvalue computations. To understand more contemporary algorithms, coverage of Krylov subspace algorithms such as CG and GMRES, as well as sparse matrix algorithms, are considered increasingly important additions to the standard canon above.

P.S. The key to numerical work is to figure out methods to solve problems for special matrices (e.g. diagonal matrices or upper triangular matrices), then figure out ways to transform entire classes of matrices into such special forms. This took me a while to appreciate consciously but apparently this is an observation that is too trivial for many textbook authors to write out explicitly. Trefethen and Bau is nice because it actually tries to motivate the development pedagogically rather than presenting a laundry list of Things You Should Know.

When is a finite matrix a "good" approximate representation of an operator?

2010-08-22T22:08:38Z

I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $\rho(r), \; r\in \mathbb{R}^3$ by a finite linear combination of basis functions

$\rho(r) = \sum_{i=1}^N q_i \phi_i (r) $

where $\phi_i (r)$ is normalized to $\int_{\mathbb{R}^3} \phi_i (r) dr = 1$ and has the interpretation of being the shape of some charge distribution (shape) of a unit charge. $\rho$ and the $\phi_i$s are real-valued but may be positive in some regions and negative in others. The basis functions are nonorthogonal and local in space but not strictly compact. Let's say for now that we use spherical Gaussians of the form $\phi_i (r) \propto \exp (-\alpha_i |r-R_i|^2)$, where $R_i$ is where the basis function is centered around. The number of basis functions chosen scales approximately as the number of atoms, as we expect charge to concentrate around atomic nuclei. (We may add additional basis function per atom of different shapes until we achieve a reasonable approximation to the desired shape of the charge distribution around an atom.)

The energy of the system can then be given by

$E = \frac 1 2 \sum_{i,j=1}^N q_i q_j J_{ij}$

where the matrix $J$ has elements

$J_{ij} = \int_{\mathbb{R}^{3\times 2}} \frac{\phi_i(r_1) \phi_j(r_2)}{|r_1 - r_2|} dr_1 dr_2$

and represents the Coulomb interaction between the unit charge distributions $\phi_i$ and $\phi_j$.

One way to look at the matrix $J$ is as a finite dimensional (approximate) representation of the Coulomb operator $\hat J = 1 / {|r_1 -r_2|}$. We know that $\hat J$ has certain nice properties such as positivity, so we expect a "good" representation of $\hat J$ should be a symmetric positive definite matrix.

My question is this: are there conditions on the discrete representation (possibly expressible as conditions on the {$\phi_i$} basis) to detect whether or not a given claimed representation $J$ is "good" in that it preserves such properties? Or asked another way, if I have some matrix $J$ which is claimed to represent $\hat J$, what are necessary and sufficient conditions on its matrix elements for it to be a "good" representation of $\hat J$?

I hope the question makes sense, and that I am not misusing too much terminology.

Answer by Jiahao Chen for Eigenvalues of A+B where A is symmetric positive definite and B is diagonal

2010-08-26T13:00:18Z

This previous MO question may be relevant.

Answer by Jiahao Chen for solving series of linear systems with diagonal perturbations

2010-08-26T11:54:16Z

a) There are formulae such as the Woodbury identity that allow for rank k updates to a previously solved problem, which I think fits your problem nicely.

b) In addition, using a reasonably smart iterative algorithm such as conjugate gradients (or whatever is appropriate for your problem) will also be helpful since you can feed it the solution from your previous problem, and for small perturbations the new solution can be computed very quickly.

In practice I have found it sufficient to use just (b), but it might be worth trying both separately or together.

Answer by Jiahao Chen for Justification for the matching condition for the wave function at potential jumps. Why is it both restrictive enough and sufficiently general?

2010-08-19T19:17:53Z

Here is a tangential response to your first question: sometimes these discontinuities do have physical significance and are not just issues of mathematical trickery surrounding pathological cases. Wavefunctions for molecular Hamiltonians become pointy where the atomic nuclei lie, which indicate places where the 1/r Coulomb operator becomes singular. There are equations like the Kato cusp conditions (T. Kato, Comm. Pure Appl. Math. 10, 151 (1957)) that relate the magnitude of the discontinuity at the nucleus to the size of the nuclear charge. I have heard this explained as a result of requiring the energy (which is the Hamiltonian's eigenvalue) to remain finite everywhere, thus at places where the potential is singular, the kinetic energy operator must also become singular at those places. Since the kinetic energy operator also controls the curvature of the wavefunction, the wavefunction at points of discontinuity must change in a nonsmooth way.

Answer by Jiahao Chen for Does there exist a potential which realizes this strange quantum mechanical system?

2010-08-19T18:52:11Z

If I understand your first question correctly, then the answer is yes. In fact, all physical matter exhibits this behavior. Allow me to answer in the following mathematically nonrigorous way:

Consider that even in a lone hydrogen atom, the Hamiltonian operator for the nonrelativistic electron

$H = - \frac 1 2 \nabla^2 + \frac{1}{r}$

has a discrete spectrum of bound states corresponding to the 1s, 2s, 2p, 3s, ... atomic orbitals and a continuous spectrum of unbound states corresponding to an electron that is unbound for all practical purposes. Thus at sufficiently high temperature (probably at $\beta^{-1}$ = kT ~ 0.5) there will be significant population of the continuous spectrum and you would have to deal with counting the continuous spectrum in the partition function.

The same phenomenon exists for all atoms and collections of atoms, even when the nuclear and interactions terms are turned on.

I am not 100% confident that the same thing holds in the relativistic case too, but I would be surprised if it did not.

Regarding your discussion of the harmonic oscillator, and the comment that "such a system [exhibiting such divergence at a critical temperature] is most likely an approximation of another system", I would go so far as to say that it is the other way round, that almost all the time "nice" systems like the harmonic oscillator are in fact derived as asympotic approximations to messier Hamiltonians. For example, you could write down the molecular Hamiltonian

$H = \sum_i -\frac 1 2 \nabla_i^2 + \sum_{ij} \frac 1 {r_{ij}} - \sum_{Ki} \frac {Z_K} {r_{iK}} + \sum_K -\frac 1 2 \nabla_K^2 $

which as mentioned above has both a discrete part and a continuous part to its spectrum, and assume that we are interested only in the regime where we care about slow atomic nuclear motions, and that they move very little, and from there derive an effective lattice Hamiltonian of coupled harmonic oscillators. While the phase transition can be observed in the original molecular Hamiltonian, it would not be possible to see this occur in the simplified Hamiltonian since the the discrete spectrum of the harmonic oscillators would go on forever without becoming continuous.

Answer by Jiahao Chen for Fundamental Examples

2009-11-21T19:21:06Z

Some people may disagree that this is an example per se, but I'd put up the Feynman path integral (see Wikipedia), because:

it provided a completely new physical picture of quantum mechanics
it led to systematic development of quantum field theory and string theories, both of which have had led to enormous synergistic growth in mathematics
it uncovered a fundamental similarity between of stochastic processes and deterministic quantum dynamics
it used the connection between Lie algebras and Lie groups in new and unexpected ways
questions about the path measure have stimulated much development in measure theory and analysis
tricks like the Wick rotation not only relate statistical mechanics and quantum mechanics (and the corresponding field theories) to each other, but also have stimulated further research in applications of analytic continuation

...and probably more that I am unaware of.

Answer by Jiahao Chen for Where does a math person go to learn statistical mechanics?

2009-11-13T19:00:49Z

A really good first textbook for statistical mechanics is David Chandler's Introduction to Modern Statistical Mechanics. It's written by a physical chemist for senior undergraduates and does an excellent job distilling down the very fundamental material into a one-semester course at Berkeley. It's not at all math heavy.

Answer by Jiahao Chen for Where does a math person go to learn statistical mechanics?

2009-11-13T18:38:37Z

Many physics people like Pathria's Statistical Mechanics. It's good for physical intuition.

Answer by Jiahao Chen for Mathematical Physics? (Particularly computational)

2009-11-13T18:33:23Z

Another book is Robert Geroch's Mathematical physics, although this may perhaps be more properly characterized as a book on modern mathematics for physicists.

Answer by Jiahao Chen for Mathematical Physics? (Particularly computational)

2009-11-13T18:23:40Z

Courant and Hilbert is great. However, it predates many significant developments in mathematics and physics in much of the 20th century. A more recent such work is Reed and Simon's Methods of Modern Mathematical Physics.

Answer by Jiahao Chen for Where does a math person go to learn quantum mechanics?

2009-11-11T03:59:46Z

I think there are some excellent recommendations above. I learned quantum mechanics for real from Shankar, I think it's a great choice. Griffiths is also a great physics text. I would also recommend these following less famous books:

Physical chemistry and materials science textbooks. I would also highly recommend newer textbooks in physical chemistry as a perhaps less obvious place to look for excellent introductions to quantum mechanics - as Dirac famously said once, it's really the foundation for all of chemistry. An excellent physical chemistry is Physical Chemistry by Berry, Rice and Ross. Presumably there are also good introductions in materials science books, although I don't have any to recommend.

Not Feynman. In my opinion Feynman's Lectures in Physics is great for insight, but it's a terrible idea to learn anything from it the first time - remember that when Feynman actually lectured, most of the freshmen and sophomores (the intended audience) dropped the course, and were replaced by senior students!

Weyl (group theory). I'm surprised no one's mentioned Hermann Weyl's textbook "Theory of groups and quantum mechanics". It's an oldie but goodie, and perhaps best appreciated with someone with a good background in group theory.

Lieb (analysis). I recommend Elliott Lieb's Analysis GSM textbook - on the surface, it looks like it's about functional analysis, but it's secretly also a text on quantum mechanics!

There are some subjects that none of the introductory quantum mechanics texts I've read ever do a satisfactory job of explaining, and I think are really worth following up after Shankar or another such book. The most important ones I think are:

Many-body phenomena. This is really where some of the strangest predictions of quantum mechanics come from, like the EPR paradox and spin statistics. Levine's Physical Chemistry is an excellent place to start. Another great book is Blaizot and Ripka's Quantum Theory of Finite Systems, which does a superb job with boson and fermion statistics.
Dynamics (time-dependent quantum mechanics. I cannot recommend Tannor's Introduction to Quantum Mechanics: A Time-dependent Perspective enough as a really fantastic resource for learning how practicing physicists and chemists actually do these calculations, beyond the really simplistic calculations presented in most introductory texts. That could also work as a first textbook.

You know, I'm in the building next to you. Maybe you should come by and talk sometime. :)

Graphs with incidence matrices whose pseudoinverses are proportional to their transposes

2009-11-10T19:41:47Z

When I was working on my PhD dissertation, I came across a physical situation involving nodes and flows between them. It turned out that I was working with a complete oriented graph $K_n$ (all nodes are connected to each other), and I needed to calculate the pseudoinverse of its incidence matrix T, i.e. the rectangular matrix N(V) x N(E) where N(E) = n is the number of edges and N(V) is the number of vertices, with matrix element 1 if the edge enters the vertex, -1 if the edge leaves the vertex, and 0 otherwise.

To my surprise the pseudoinverse turns out to be proportional to the transpose of the incidence matrix! Specifically

$T^{+} (K_n) = \frac{1}{n} T^{\prime} (K_n)$

where the prime denotes transposition.

My question is:

A) What other graphs $G$, if any, have this property, i.e. that

$T^+(G) \propto T^\prime(G)$,

or some suitable generalization thereof, and

B) How can I show that this result is invariant of orientation? (I determined empirically is certainly true for all possible orientations of the small complete graphs, and I haven't been able to find a counterexample, but I don't have a proof of this statement yet)

I'm not a professional mathematician, so any thoughts would be welcome.

One class of generalizations that is possible (but not so interesting IMO) are disconnected graphs where each subgraph is a complete graph, i.e.

$G = K_{n_1} \oplus K_{n_2} \oplus \dots \oplus K_{n_m}$

In this case one gets

$T^+(G) = \frac{1}{n_1} T^\prime(K_{n_1}) \oplus \frac{1}{n_2} T^\prime(K_{n_2}) \oplus \dots \oplus \frac{1}{n_m} T^\prime(K_{n_m})$

which is not really that exciting, but perhaps could point the way to a more interesting generalization.

P.S. There is a short proof of the first fact, which relies on the fact that the complete graph has a Laplacian of the form

$\Delta = n \mathbf{I} - \mathbf{1} \mathbf{1}^\prime$

where $\mathbf{I}$ is the n x n identity matrix and $\mathbf{1}$ is a column vector of ones.

With this fact, together with knowing that the column sums of $T$ are all zero, it is straightforward to show that $T^\prime(K_n)/n$ satifies all the Moore-Penrose conditions for the pseudoinverse.

P.P.S. If anyone is interested in the physical context, here is where it came from.

Answer by Jiahao Chen for Which computer algebra system should I be using to solve large systems of sparse linear equations over a number field?

2009-11-10T19:08:38Z

A system that large is large enough where the speedup from skipping over all the layers of abstraction in most CAS packages is worth the trouble to write your own custom code to solve the problem.

If you are ok with a floating point solution, ScaLAPACK or another linear algebra package that has algorithms and data structures for sparse matrices would be a lot better than (P)LAPACK, which as far as I know uses only dense matrix data structures.

Python's scipy package also has sparse data structures and wrappers to call UMFPACK, and the syntax is easy enough that it wouldn't be significantly harder to use than a CAS program. It would be easier than writing a custom Fortran/C program straight up.

For solutions over the reals, there is an algorithm called the complete orthogonal decomposition (COD) that uses rank revealing QR factorization (see, e.g. Golub's Matrix Computations). This lets you separate out and project away the kernel of the problem, leaving behind only the part of the problem that lies in the range. I don't know if there is an analogue for arbitrary fields, but since you appear to have a problem with a small rank, it may be worth your trouble to look into this.

Comment by Jiahao Chen

2012-06-08T05:42:16Z

This is actually not too different from Sheehan Olver's published method.

Comment by Jiahao Chen

2012-05-25T03:30:33Z

I found this to be a nice introduction to the topic. ams.org/notices/200102/fea-knutson.pdf

Comment by Jiahao Chen

2012-05-22T06:11:22Z

@Gjergji my apologies for the duplication. FWIW I did try to search for this question but I didn't quite turn up anything on my own.

Comment by Jiahao Chen

2012-05-22T06:05:20Z

@Joseph - thank you, I had the feeling that I was missing some magic incantation to find the relevant literature.

Comment by Jiahao Chen

2012-05-14T01:31:46Z

To be honest, I was expecting to hear "this is obvious and well known and the answer is X", but it certainly seems like this is a potential research area of wide interest.

Comment by Jiahao Chen

2012-05-14T01:25:06Z

Thinking a bit more about b), this suggests that the norm of $\log \left(\exp A \exp B\right) - (A + B)$ might be an interesting object to look at

Comment by Jiahao Chen

2012-05-14T01:21:29Z

After thinking about this a little bit, it seems rather interesting that the Schatten norm, which contains information about eigenvalues of the commutator, is used to quantify something which one would naively associate with the eigenvectors/eigenfunctions that define the relative basis between $A$ and $B$. I'm not surprised that at least some information about the basis vectors/functions is retained in the Schatten norm, but I wonder how much is known about this as well.

Comment by Jiahao Chen

2012-05-13T13:22:09Z

Why am I not at all surprised that Catalan numbers showed up here? This is a most illuminating paper. Thank you!

Comment by Jiahao Chen

2012-05-13T01:46:05Z

The right hand side is just $ \binom {k+z} k $ or its suitable generalization to noninteger $z$, for what it's worth.

Comment by Jiahao Chen

2012-05-12T08:40:42Z

I would agree that b) is the fairly straightforward generalization of what I had in mind, but I'm holding out to see if someone has a smarter way of doing it beyond checking each term which seems like a test "by brute force".

Comment by Jiahao Chen

2012-05-12T01:52:52Z

It seems like the Schatten norm should grow something like ~O(n) for n-by-n finite-dimensional matrices, assuming that eigenvalues don't vary with n? It might even be interesting to see if there is an onset of such behavior even in finite-rank approximations to noncommutative operators.

Comment by Jiahao Chen

2012-05-12T01:43:26Z

Thanks! This is certainly an interesting fine-grained look at the question of quantifying noncommutativity.

Comment by Jiahao Chen

2012-05-11T19:57:20Z

Perhaps a sentence got cut off? 'the values of p for which the Schatten p-norm of the commutator of two operators [...] is a good measure of non-commutativity.'

Comment by Jiahao Chen

2012-05-11T18:25:25Z

Certainly; done.

Comment by Jiahao Chen

2012-05-11T18:01:48Z

I would agree that there is something special about the matrix structure that allows this correspondence. I was not aware of Heine's formula, thanks. This deserves to be accepted as an answer, although I wonder if there is something more interesting that points toward possible generalizations.