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

President Grover Cleveland’s Secret Surgery on the Steam Yacht Oneida–Magic Masts and Sturdy Ships


The President stood at the rail of his friend’s yacht, the Oneida, watching the waves from Long Island Sound roll and tumble over each other. His fingers itched for his fishing rod. He had fished from this yacht many times in the past, but this time was different. This time, he faced something more serious than how many fish he caught. His tongue explored the contours of the tumor growing on the roof of his mouth. The economic panic threatened the country like his tumor threatened his mouth. He didn’t want to call it cancer. Cancer, the forbidden word that translated into a person just as forbidden. The operation to remove the growth from his mouth had to remain secret for the good of the country and for the good of his family.

From Magic Masts and Sturdy Ships

Too bad he didn’t choose one of George Warrington’s steam yachts, but alas the Warringtons (and many of Chicago’s grandees) were good Republicans…but that would pay off when Theodore Roosevelt appointed him as Engineering Commissioner of Lighthouses and Lightships a decade later.

This Vessel Served in the U.S. Coast Guard in Peacetime and in the United Kingdom’s Royal Navy During World War II — Transportation History

October 2, 1930 USCGC Saranac, one of the Lake-class cutters of the U.S. Coast Guard (USCG), was officially commissioned as a vessel of that military branch. This cutter had been launched in April of that year at the yards of the General Engineering and Drydock Company in Oakland, California. USCG Captain John Boedker oversaw the […]

This Vessel Served in the U.S. Coast Guard in Peacetime and in the United Kingdom’s Royal Navy During World War II — Transportation History

An Aviation Legend Makes His Most Pivotal Contribution to Flight Technology — Transportation History

September 24, 1929 U.S. Army Air Corps (USAAC) Lieutenant James H. “Jimmy” Doolittle, who would achieve lasting fame as commander of the Doolittle Raid during World War II, made his most significant contribution to aeronautical technology when he guided a Consolidated N-Y-2 Husky training biplane over Mitchel Field in New York in what was the […]

An Aviation Legend Makes His Most Pivotal Contribution to Flight Technology — Transportation History

Jimmy Doolittle figures in Chet’s aviation story too as a competitor in the first Washington Air Derby (later Langley Day) held in 1932. Chet was the first president of the Washington Air Derby Association.