Specht's theorem

Combinatorics on words | Matrix theory | Theorems in linear algebra

In mathematics, Specht's theorem gives a necessary and sufficient condition for two complex matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940. Two matrices A and B with complex number entries are said to be unitarily equivalent if there exists a unitary matrix U such that B = U *AU. Two matrices which are unitarily equivalent are also similar. Two similar matrices represent the same linear map, but with respect to a different basis; unitary equivalence corresponds to a change from an orthonormal basis to another orthonormal basis. If A and B are unitarily equivalent, then tr AA* = tr BB*, where tr denotes the trace (in other words, the Frobenius norm is a unitary invariant). This follows from the cyclic invariance of the trace: if B = U *AU, then tr BB* = tr U *AUU *A*U = tr AUU *A*UU * = tr AA*, where the second equality is cyclic invariance. Thus, tr AA* = tr BB* is a necessary condition for unitary equivalence, but it is not sufficient. Specht's theorem gives infinitely many necessary conditions which together are also sufficient. The formulation of the theorem uses the following definition. A word in two variables, say x and y, is an expression of the form where m1, n1, m2, n2, …, mp are non-negative integers. The degree of this word is Specht's theorem: Two matrices A and B are unitarily equivalent if and only if tr W(A, A*) = tr W(B, B*) for all words W. The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset. Let n denote the size of the matrices A and B. For the case n = 2, the following three conditions are sufficient: For n = 3, the following seven conditions are sufficient: For general n, it suffices to show that tr W(A, A*) = tr W(B, B*) for all words of degree at most It has been conjectured that this can be reduced to an expression linear in n. (Wikipedia).

Calculus 5.3 The Fundamental Theorem of Calculus

My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart

From playlist Calculus

The Divergence Theorem

Divergence Theorem. In this video, I give an example of the divergence theorem, also known as the Gauss-Green theorem, which helps us simplify surface integrals tremendously. It's, in my opinion, the most important theorem in multivariable calculus. It is also extremely useful in physics,

From playlist Vector Calculus

Introduction to additive combinatorics lecture 10.8 --- A weak form of Freiman's theorem

In this short video I explain how the proof of Freiman's theorem for subsets of Z differs from the proof given earlier for subsets of F_p^N. The answer is not very much: the main differences are due to the fact that cyclic groups of prime order do not have lots of subgroups, so one has to

From playlist Introduction to Additive Combinatorics (Cambridge Part III course)

Introduction to additive combinatorics lecture 1.8 --- Plünnecke's theorem

In this video I present a proof of Plünnecke's theorem due to George Petridis, which also uses some arguments of Imre Ruzsa. Plünnecke's theorem is a very useful tool in additive combinatorics, which implies that if A is a set of integers such that |A+A| is at most C|A|, then for any pair

From playlist Introduction to Additive Combinatorics (Cambridge Part III course)

Calculus - The Fundamental Theorem, Part 1

The Fundamental Theorem of Calculus. First video in a short series on the topic. The theorem is stated and two simple examples are worked.

From playlist Calculus - The Fundamental Theorem of Calculus

Ultra low-temperature cascade refrigeration system repair

I bought a -100*C refrigerated cold trap from a surplus sale, which was not operational. In this video, I describe how the unit works after recharging its second stage with R1150 (ethylene) refrigerant. Savant RVT4104 - (manual) http://phoenix.tuwien.ac.at/chemistry/Gebrauchsanweisungen/

From playlist Mechanics

Multivariable Calculus | The Squeeze Theorem

We calculate a limit using a multivariable version of the squeeze theorem. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Multivariable Calculus

The Topology of Restricted Partition Posets - Richard Ehrenborg

Richard Ehrenborg University of Kentucky; Member, School of Mathematics November 2, 2010 The d-divisible partition lattice is the collection of all partitions of an n-element set where each block size is divisible by d. Stanley showed that the Mobius function of the d-divisible partition

From playlist Mathematics

EEVblog #1164 - Xbox Engineering Baptism Of Fire

A retro look at the 2005 Xbox 14 million unit power cord replacement issue and a teardown and analysis of the supplied protection device from Microsoft. Some interesting engineering lessons to be learned. UPDATE: Apparently what this is an "Arc Fault Circuit Interrupter" (AFCI) that contin

From playlist Electronics Design / Build

Divergence theorem of Gauss

Free ebook http://tinyurl.com/EngMath A short tutorial on how to apply Gauss' Divergence Theorem, which is one of the fundamental results of vector calculus. The theorem is stated and we apply it to a simple example.

From playlist Several Variable Calculus / Vector Calculus

Tangent Lines to sin(x)

Calculus: Find all x such that the tangent line to f(x) = sin(x) has slope equal to sqrt(3)/2 and -sqrt(2)/2.

From playlist Calculus Pt 1: Limits and Derivatives

Some new results on rationality (Lecture - 01) by Claire Voisin

Infosys-ICTS Ramanujan Lectures Some new results on rationality Speaker: Claire Voisin (College de France) Date: 01 October 2018, 16:00 Venue: Madhava Lecture Hall, ICTS campus Resources Lecture 1: Some new results on rationality Date & Time: Monday, 1 October 2018, 04:00 PM Abstra

From playlist Infosys-ICTS Ramanujan Lectures

Calculus - The Fundamental Theorem, Part 3

The Fundamental Theorem of Calculus. Specific examples of simple functions, and how the antiderivative of these functions relates to the area under the graph.

From playlist Calculus - The Fundamental Theorem of Calculus

Coherent (phi, Gamma)-modules and cohomology of local systems by Kiran Kedlaya

PERFECTOID SPACES ORGANIZERS : Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri and Narasimha Kumar Cheraku DATE & TIME : 09 September 2019 to 20 September 2019 VENUE : Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknat

From playlist Perfectoid Spaces 2019

Stokes Theorem

In this video, I present another example of Stokes theorem, this time using it to calculate the line integral of a vector field. It is a very useful theorem that arises a lot in physics, for example in Maxwell's equations. Other Stokes Example: https://youtu.be/-fYbBSiqvUw Yet another Sto

From playlist Vector Calculus

A gentle introduction to group representation theory -Peter Buergisser

Optimization, Complexity and Invariant Theory Topic: A gentle introduction to group representation theory Speaker: Peter Buergisser Affiliation: Technical University of Berlin Date: June 4, 2018 For more videos, please visit http://video.ias.edu

From playlist Mathematics

Contract Law 33 II Sprecht v Netscape (no “clickwrap”)

II. THE BARGAIN RELATIONSHIP E. Nature and Effect of Counter Offer Sprecht v. Netscape, (no “clickwrap”) To access case file, copy and paste link into browser - ianayres.com/sites/default/files/files/Specht%20v_%20Netscape.docx These video lectures are taken from Prof. Ayres’ Coursera

From playlist American Contract Law

When Time Became History - The Human Era

Get Merch designed with ❤ from https://kgs.link/shop-131 Join the Patreon Bird Army 🐧 https://kgs.link/patreon ▼▼ More infos and links are just a click away ▼▼ For the fifth time, we present to you the Human Era Calendar for the year 12,021 – this time it is all about the journey of h

From playlist Medicine & Biology

Is It Too Late To Stop Climate Change? Well, it's Complicated.

Head over to our shop to get exclusive kurzgesagt merch and sciency products designed with love. Getting something from the kurzgesagt shop is the best way to support us and to keep our videos free for everyone. ►► https://kgs.link/shop-130 (Worldwide Shipping Available) A special thank

From playlist The Existential Crisis Playlist

C73 Introducing the theorem of Frobenius

The theorem of Frobenius allows us to calculate a solution around a regular singular point.

From playlist Differential Equations

Specht's theorem

Related pages