[permalink] [id link]
the matrices satisfy various polynomials such as their minimal polynomials, which form a proper ideal ( because they are not all zero, in which case the result is trivial ); one might call this the characteristic ideal, by analogy with the characteristic polynomial.
from
Wikipedia
Some Related Sentences
matrices and satisfy
A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation:
It might be tempting to suppose a matrix with orthogonal ( not orthonormal ) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name ; they only satisfy M < sup > T </ sup > M = D, with D a diagonal matrix.
Analogous to numbers ( elements of a field ), matrices satisfy the following general properties.
The Lie algebra of Sp ( 2n, F ) is given by the set of 2n × 2n matrices A ( with entries in F ) that satisfy
The Lie algebra of Sp ( n ) is given by the quaternionic skew-Hermitian matrices, the set of n by n quaternionic matrices that satisfy
Specht's theorem states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities.
Nevertheless, as matrices, X ( t ) and P ( t ) satisfy the classical equations of motion.
Planck's constant does not appear in the classical equations, so that the matrices could be constructed for many different values of ħ and still satisfy the equations of motion, but with different energy levels.
Fortunately, matrices which satisfy the conditions to be used in the Hill cipher are fairly common.
The half matrices satisfy that
where V < sub > k </ sub > are called Kraus matrices which satisfy
Additionally, in the case of square matrices ( thus, m = n ), some ( but not all ) matrix norms satisfy the following condition, which is related to the fact that matrices are more than just vectors:
The definition of an association scheme is equivalent to saying that the are v × v ( 0, 1 )- matrices which satisfy
matrices and various
When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors.
These differences are caused by the different lipophilicity of the various side chains, and by the different food matrices in which they occur.
The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function ( see below ).
Since the off-diagonal elements-the coherence terms-decay for increasing, then the density matrices for the various qubits of the system will be indistinguishable.
Well-known families of Pauli matrices, Gell-Mann matrices and various generalizations are Hermitian.
Systems of equations may also be represented in terms of matrices, allowing various principles of matrix operations to be handily applied to the problem.
The various factors in this expression are gamma matrices as in the covariant formulation of the Dirac equation ; they have to do with the spin of the electron.
Kneeboard designers however are known for their wild experimental excess and so most modern materials including various aerospace elements such as Titanium alloys ( for fins ), carbon fibre and kevlar in epoxy matrices are not unusual.
Specifically, applying the structure theorem for finitely generated modules over a principal ideal domain to this algebra yields as corollaries the various canonical forms of matrices, such as Jordan canonical form.
It collects the various partial derivatives of a single function with respect to many variables, and / or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
matrices and polynomials
He was one of the foremost analysts of his generation and made fundamental contributions to the theory of Toeplitz matrices and orthogonal polynomials.
He was one of the foremost analysts of his generation and made fundamental contributions to the theory of Toeplitz matrices and orthogonal polynomials.
The simplest proofs use just those notions needed to formulate the theorem ( matrices, polynomials with numeric entries, determinants ), but involve technical computations that render somewhat mysterious the fact that they lead precisely to the correct conclusion.
It is possible to avoid such details, but at the price of involving more subtle algebraic notions: polynomials with coefficients in a non-commutative ring, or matrices with unusual kinds of entries.
This is important to note here, because these relations will be applied for matrices with non-numeric entries such as polynomials.
This proof uses just the kind of objects needed to formulate the Cayley – Hamilton theorem: matrices with polynomials as entries.
For summations defined in terms of addition of other values than numbers ( such as vectors, matrices, polynomials ), in general of values in some given Abelian group, the value of an empty summation is taken to be the zero element of that group.
If A and B are two square n × n matrices then characteristic polynomials of AB and BA coincide:
Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group ( or even monoid ).
This debate was due to the confusion of quadratic forms ( represented by polynomials ) and symmetric bilinear forms ( represented by matrices ), and " twos out " is now the accepted convention ; " twos in " is instead the theory of integral symmetric bilinear forms ( integral symmetric matrices ).
The determinant is over the ring of n × n matrices whose entries are polynomials in t with coefficients in the commutative algebra of even complex differential forms on M. The curvature form of V is defined as
Topics include polynomials, algebraic equations, separation of roots including Sturm's theorem, approximation of roots, and the application of matrices and determinants to the solving of equations.
The coefficients of the character formulas for infinite dimensional irreducible representations of E < sub > 8 </ sub > depend on some large square matrices consisting of polynomials, the Lusztig – Vogan polynomials, an analogue of Kazhdan – Lusztig polynomials introduced for reductive groups in general by George Lusztig and David Kazhdan ( 1983 ).
The values at 1 of the Lusztig – Vogan polynomials give the coefficients of the matrices relating the standard representations ( whose characters are easy to describe ) with the irreducible representations.
Symbolically, it can do multivariate polynomial arithmetic, factor polynomials, compute GCDs, expand series, and compute with matrices.
The notion of stability radius is generally applied to special functions as polynomials ( the spectrum is then the roots ) and matrices ( the spectrum is the eigenvalues ).
These matrices are used in commutative algebra, e. g. to test if two polynomials have a ( non constant ) common factor.
matrices and such
Compost is generally recommended as an additive to soil, or other matrices such as coir and peat, as a tilth improver, supplying humus and nutrients.
* Advanced composite ceramic and metal matrices have been designed for most modern armoured fighting vehicles because they offer superior penetrating resistance against shaped charges such as high explosive antitank ( HEAT ) rounds and kinetic energy penetrators.
Dyson also did work in a variety of topics in mathematics, such as topology, analysis, number theory and random matrices.
* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.
* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.
Multiplication is also defined for other types of numbers ( such as complex numbers ), and for more abstract constructs such as matrices.
For example, methods based on covariance matrices are typically employed on the premise that numbers, such as raw scores derived from assessments, are measurements.
However, one can consider arbitrary types such as boolean values, categorical variables, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, functions, and processes.
The non-degeneracy condition means that for all we have the property that there does not exist non-zero such that for all The skew-symmetric condition ( inherent in the definition of differential 2-form ) means that for all we have for all Recall that in odd dimensions antisymmetric matrices are not invertible.
Fix a set of matrices such that ( i. e. fix a convention for the gamma matrices ).
One needs to construct such matrices explicitly, however.
Pairwise counts are often displayed in matrices such as those below.
Often for reasons of geometric interest this is specialized to a subcategory, by requiring that the nondegenerate bilinear forms have additional properties, such as being symmetric ( orthogonal matrices ), symmetric and positive definite ( inner product space ), symmetric sesquilinear ( Hermitian spaces ), skew-symmetric and totally isotropic ( symplectic vector space ), etc.
The primary on-board high-level programming languages of most graphing calculators ( most often Basic variants, sometimes Lisp derivatives, and more uncommonly, C derivatives ) in many cases can glue together calculator functions — such as graphs, lists, matrices, etc.
For this reason, fuzzy logic usually uses IF-THEN rules, or constructs that are equivalent, such as fuzzy associative matrices.
The term " sedenion " is also used for other 16-dimensional algebraic structures, such as a tensor product of 2 copies of the quaternions, or the algebra of 4 by 4 matrices over the reals, or that studied by.
SVD of matrix A computes unitary matrices U and V and a rectangular diagonal matrix S of the same size as A with nonnegative diagonal entries, such that
Computation of the SVD of a matrix generally costs about the same as several matrix-matrix multiplications with matrices of the same size when state-of-the art implementation ( accurate up to rounding precision ) is used, such as in LAPACK.
This concept makes sense, however: such matrices have no entries and so are completely determined by their size.
Thinking about such matrices can be useful in one way, though: they highlight the fact that given any objects and in an additive category, there is exactly one morphism from to 0 ( just as there is exactly one 0-by-1 matrix with entries in ) and exactly one morphism from 0 to ( just as there is exactly one 1-by-0 matrix with entries in ) – this is just what it means to say that 0 is a zero object.
( In this case, ' large ' means, approximately, more than 100, 000 elements in each matrix, or enough addressable memory such that the matrices will not fit in L1 and L2 caches.
Several silanones have been studied in argon matrices and in the gas phase, but they are highly reactive .< ref > Despite their reactivity, silanones are important as intermediates in gas-phase processes such as chemical vapor deposition in microelectronics production, in the formation of ceramics by combustion ,< ref > and in astrochemistry.
( In post-Crisis on Infinite Earths continuity, the Eradicator, an artificial life form programmed to preserve all Kryptonian culture, altered the birthing matrices (" artificial wombs ") that the explorers took with them so that all newborns would be fatally vulnerable to lead and other materials such as greenhouse gases and certain rocks.
0.923 seconds.