Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra

Theorem 1 (Eigenvector-eigenvalue identity) Let {A} be an {n \times n} Hermitian matrix, with eigenvalues {\lambda_1(A),\dots,\lambda_n(A)}. Let {v_i} be a unit eigenvector corresponding to the eigenvalue {\lambda_i(A)}, and let {v_{i,j}} be the {j^{th}} component of {v_i}. Then

\displaystyle |v_{i,j}|^2 \prod_{k=1; k \neq i}^n (\lambda_i(A) - \lambda_k(A)) = \prod_{k=1}^{n-1} (\lambda_i(A) - \lambda_k(M_j))

where {M_j} is the {n-1 \times n-1} Hermitian matrix formed by deleting the {j^{th}} row and column from {A}.

When we posted the first version of this paper, we were unaware of previous appearances of this identity in the literature; a related identity had been used by Erdos-Schlein-Yau and by myself and Van Vu for applications to random matrix theory, but to our knowledge this specific identity appeared to be new. Even two months after our preprint first appeared on the arXiv in August, we had only learned of one other place in the literature where the identity showed up (by Forrester and Zhang, who also cite an earlier paper of Baryshnikov).

Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv a completely rewritten version of our previous paper, now titled “Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra“. This paper is now a survey of the various literature surrounding the following basic identity in linear algebra, which we propose to call the eigenvector-eigenvalue identity:

Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra

The Slow Suicide of American Science–ACSH

I’ve always been bullish about American scientific and technological supremacy, not in some starry-eyed, jingoistic way, but due to the simple reality that the United States remains the world’s research and development engine.

This is true for at least four reasons, which I detailed previously: (1) Superior higher education; (2) A cultural attitude that encourages innovation; (3) Substantial funding and financial incentives; and (4) A legal framework that protects intellectual property and tolerates failure through efficient bankruptcy laws. There’s a fifth, fuzzier reason, namely that smart and talented people have long gravitated toward the U.S.

The Slow Suicide of American Science–ACSH

Don’t Try to Predict Physics (or Much of Anything Else) Without a Model

A salutary note at the end of Rutherford Aris’ Mathematical Modelling Techniques:

When a model is being used as a simulation an obvious comparison can be made between its predictions and the results of the experiment. We are favourably impressed with the model if the agreement is good and if it has not been purchased at the price of too many empirical constant adjusted to fit the data. If the parameters are determined independently and fed into the final model as fixed constant not to be further adjusted, then we can have a fair degree of confidence in the data and in the model. Both model and data have their own integrity the former in the relevance and clarity of its hypotheses and the rigour and appropriateness of its development, the latter in the carefulness of the experimenter and the accuracy of the results. But these virtues do not only inhere in the possessors they also gain validity from the other…Thus the attitude of never believing an experiment until its confirmed by theory has as much to be said for it as that which never believes a theory before its confirmation by experiment. (emphasis mine)

In the comparison of theory with experiment an array of statistical tools is available and should be used. One danger that is easy to overlook is the existence of hidden constancies that will give spurious values…The classic correlation between the intelligence of the children and the drunkenness of the parents which so confounded temperance societies years ago–until it was discovered that all the data came from schools in the east end of London–is another illustration of a data base too narrow to test a model.

As someone who works in the earth sciences, the indiscriminate use of statistics and purely empirical relationships is maddening, and that has spread to many other disciplines as well. The computer power we have at our disposal these days makes it too tempting to simply reduce “big data” and let us “tell us” what’s going on, but this can be a serious mistake without some kind of hypothesis–right or wrong–about what we are looking at.

Chauvenet’s Criterion

In the pressure gauge testing lab experiment, one of the requirements is that the “outliers” in the data are determined and excised from the analysis. One way of doing that is to apply Chauvenet’s Criterion. Below is a video of how that’s done and who Chauvenet was.

Note: he butchers the pronunciation of Chauvenet, sorry.

Flexible Meshing Enables Accurate CFD for Nuclear Reactor Rod Bundles — Another Fine Mesh

Accurate fluid flow modeling of nuclear reactor rod bundles is essential and extremely challenging with exacting standards for the mesh to deal with the geometric complexity and near-wall physics. The tightly packed rods with wrapped wires, mainly used in liquid metal cooled systems, that contact the rods provide a challenging geometry into which well-defined boundary […]

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