mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a trace identity is any

equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...

involving the

trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...

of a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

Properties

Trace identities are invariant under simultaneous

conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the chang ...

Uses

They are frequently used in the

invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...

n \times n

matrices to find the generators and relations of the

ring of invariants In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is, :R^f = \. More generally, if ''G'' is a group acting on ''R'', then the subring of ''R'' :R^G = \ is called the ...

, and therefore are useful in answering questions similar to that posed by

Hilbert's fourteenth problem In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. The setting is as follows: Assume that ''k'' is a field and let ''K'' be a subfield of ...

Examples

* The

Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...

says that every square matrix satisfies its own

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...

. This also implies that all square matrices satisfy

\operatorname\left(A^n\right) - c_\operatorname(A) \operatorname\left(A^\right) + \cdots + (-1)^n n \det(A) = 0\,

where the coefficients

c_i

are given by the

elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sym ...

s of the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

s of . * All square matrices satisfy

\operatorname(A) = \operatorname\left(A^\mathsf\right).\,

References

{{citation, title=Graduate Algebra: Noncommutative View, volume=2, series=

Graduate Studies in Mathematics Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General To ...

, first=Louis Halle, last=Rowen, publisher=American Mathematical Society, year=2008, isbn=9780821841532, page=412, url=https://books.google.com/books?id=8svFC09gGeMC&pg=PA412. Invariant theory Linear algebra

Properties

Uses

Examples

See also

References