Non-commutative standard polynomials applied to matrices
نویسنده
چکیده
The Amitsur–Levitski Theorem tells us that the standard polynomial in 2n non-commuting indeterminates vanishes identically over the matrix algebra Mn(K). For K = R or C and 2≤ r ≤ 2n−1, we investigate how big Sr(A1, . . . ,Ar) can be when A1, . . . ,Ar belong to the unit ball. We privilegiate the Frobenius norm, for which the case r = 2 was solved recently by several authors. Our main result is a closed formula for the expectation of the square norm. We also describe the image of the unit ball when r = 2 or 3 and n = 2. MSC classification : 15A24, 15A27, 15A60
منابع مشابه
\Positive" Non-Commutative Polynomials are Sums of Squares
Hilbert's 17th problem concerns expressing polynomials on R as a sum of squares. It is well known that many positive polynomials are not sums of squares; see [R00] [deA preprt] for excellent surveys. In this paper we consider symmetric non-commutative polynomials and call one \matrix positive", if whenever matrices of any size are substituted for the variables in the polynomial the matrix value...
متن کاملOperational matrices with respect to Hermite polynomials and their applications in solving linear differential equations with variable coefficients
In this paper, a new and efficient approach is applied for numerical approximation of the linear differential equations with variable coeffcients based on operational matrices with respect to Hermite polynomials. Explicit formulae which express the Hermite expansion coeffcients for the moments of derivatives of any differentiable function in terms of the original expansion coefficients of the f...
متن کاملThe Strong Asymptotic Freeness of Haar and Deterministic Matrices
In this paper, we are interested in sequences of q-tuple of N × N random matrices having a strong limiting distribution (i.e. given any non-commutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having this property, and we show that this property pertains if the q-tuple is enlarged with independent unitar...
متن کاملNon-commutative Rational Functions in Strongly Convergent Random Variables
Random matrices like GUE, GOE and GSE have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and Thorbørnsen. It is called strong convergence property and then more random matrices with this property are followed. In general, the definition can be stated for a sequence of tuples over some C∗-algebras. In t...
متن کاملGenerating Matrix Identities and Proof Complexity Lower Bounds
Motivated by the fundamental lower bounds questions in proof complexity, we investigate the complexity of generating identities of matrix rings, and related problems. Specifically, for a field F let A be a non-commutative (associative) F-algebra (e.g., the algebra Matd(F) of d × d matrices over F). We say that a non-commutative polynomial f(x1, . . . , xn) over F is an identity of A, if for all...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2017