﻿ "Scalar" related terms, short phrases and links
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  Encyclopedia of Keywords > Mathematics > Algebra > Linear Algebra > Vectors > Scalar   Michael Charnine

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  1. A scalar is a variable that only has magnitude, e.g.
  2. A scalar is a 1-by-1 matrix; a vector is a 1-by-N or N-by-1 matrix.
  3. The scalar is called an eigenvalue of.
  4. A scalar is any real (later complex) number.
  5. A scalar is a matrix with a single row and column.


  1. The word scalar derives from the English word "scale" for a range of numbers, which in turn is derived from scala ( Latin for "ladder").
  2. The characteristic equation of a (square) matrix A is the polynomial equation: det ( A - lambda * I ) = 0 where lambda is an unknown scalar value.
  3. A scalar can be subtracted from a matrix of any size.
  4. Exhausting the above theorem the energy potential reduces to a function of 3 scalar parameters rather than 6.
  5. It must "transform" like a scalar, which really means it must not change at all under rotations and parity.

Scalar Multiplication

  2. Juxtaposition indicates either scalar multiplication or the multiplication operation in the field.
  3. A geometric interpretation to scalar multiplication is a stretching or shrinking of a vector.
  4. In other words, an algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the field.
  5. Matrix operations: addition, scalar multiplication, multiplication, transpose, inverse, and expressions involving these operations.

Scalar Field

  1. In scalar theories of gravitation scalar fields are used to describe the gravitational field.
  3. Since the Higgs field is a scalar field, the Higgs boson has no spin and has no intrinsic angular momentum.
  4. The vacuum, as scalar field, is also the rigorous identity of a massless charged field, of--for example--the electrostatic scalar potential, 0.
  5. In quantum field theory, a tachyon is a quantum of a field—usually a scalar field —whose squared mass is negative.


  2. Technically, the scalar triple product isn't a scalar, it is a pseudoscalar: under a coordinate inversion ( x goes to - x), it flips sign.
  3. It differs from the dot product in that it results in a pseudovector rather than in a scalar.
  4. Given two vectors, their scalar product is defined as the product of their magnitudes times the cosine of the angle between them.
  5. One type, the dot product, is a scalar product; the result of the dot product of two vectors is a scalar.


  1. When V is K n, then scalar multiplication is defined component-wise.
  2. Finally, addition and scalar multiplication on Fn are connected by distributive properties.
  3. De--ne scalar multiplication by a(aij ) = (aaij ) for a in F , and de--ne matrix addition by (aij ) + (bij ) = (aij + bij ).


  1. Vectors A scalar is a quantity like mass or temperature that only has a magnitude.vector is a mathematical object that has magnitude and direction.
  3. A field (scalar or vector) may be defined mathematically as some function of that vector which connects an arbitrary origin to a general point in space.

Dot Product

  2. Those abstract vectors can be added and multiplied by a scalar, and an analogue of the dot product is defined for them.
  3. Unlike the cross product, the dot product is a scalar, not a vector, and has no direction.


  1. The quantum vacuum is a dynamic massless scalar field.
  2. Through scalars and scalar fields, the virtual vacuum plasma ("virtual particle vacuum ether").
  3. In scalar EM, the disintegrated energy of vacuum is termed ANENERGY. Virtual energy is disintegrated, observable energy is integrated.

Scalar Product

  1. If the scalar product vanishes (and if neither vector in the product is the null vector) then the two vectors are orthogonal.
  2. In[1]:= Out[1]= The "dot" operator gives the scalar product of two vectors.
  4. This formula does not explicitly depend on the definition of the scalar product.
  5. The scalar product is defined for two vector operands with the result being a scalar.


  1. A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient.
  2. In classical electrodynamics, charge density -- is a scalar, the electric field, E, and current j are vectors, but the magnetic field, H is an axial vector.
  3. The terms scalar and vector as used here include pseudoscalars and pseudovectors or axial vectors (see also below).


  1. In physics, magnitude is a scalar in the physical sense, i.e.
  2. In physics, a pseudoscalar denotes a physical quantity analogous to a scalar.

Vector Field

  1. Any linear functional is either trivial (equal to 0 everywhere) or surjective onto the scalar field.
  2. The norm is usually defined to be an element of V 's scalar field K, which restricts the latter to fields that support the notion of sign.
  3. See Scalar multiplication above for a counter-example such as the scalar field of quaternions.
  4. The divergence and curl are two operations on a vector field which result in a scalar field and another vector field respectively.
  5. The anomalous interaction of the scalar, pseudoscalar, vector, and pseudovector fields with the external electromagnetic field is considered.


  1. A Lorentz scalar is generated from vectors and tensors.
  2. The inner product of the acceleration and the velocity is a Lorentz scalar and is zero.


  1. A vector space equipped with a scalar product is called a inner product space.
  2. Alternatively, a vector space V can be equipped with a norm function that assigns to every vector v in V a scalar || v ||.
  3. This text describes the multiobjective search techniques are not equipped to handle irradiation psychotronic scalar.


  2. Now consider any linear (V:h|V) and u in V with h(u) = k.u for some scalar k; unless u is an additive identity, a.k.a.

Scalar Quantity

  2. Distance is a scalar quantity, containing only a magnitude, whereas displacement is an equivalent vector quantity containing both magnitude and direction.
  3. If we form a scalar product of a vector field with the operator, we get a scalar quantity called the divergence of the vector field.
  4. It is an entirely commutative product that returns a scalar quantity.


  1. Ok. I assume that V has a scalar product <.,.
  2. Applications of multiobjective problems since they inherently assume a single scalar objective function.


  1. The operation of multiplying a vector by a scalar is called scalar multiplication.
  2. Scalar multiplication consists of multiplying each element of a matrix by a given scalar.
  3. Finally - we can multiply a scalar times a vector - multiplying a scalar s by a vector A = [x,y,z] gives [sx,sy,sz].
  5. The covariance matrix is formed by multiplying each of the M frame buffer vectors by every offer frame buffer vector (taking the scalar product).


  2. Also known as the Tesla wave, this standing scalar wave can be seen alternately as a time wave or a gravitational wave.


  1. To define vector space over the ring of scalars, we define the following two operations: addition of vectors and multiplication of vector by scalar.
  2. The vector product (the cross product) We've just seen that the scalar product (or dot product) of two vectors was a scalar.
  3. Vectors: Addition of vectors, scalar product, cross product, scalar and vector triple products.
  4. The remainder of the vector-scalar linear algebra subprograms are commonly used computations provided for your applications.


  1. The function which satisfies axioms 1)–5) is called the scalar (or inner) product of and.

Killing Form

  1. Returns the Killing Form of two elements (scalar product).
  2. If g is a simple Lie algebra then any invariant symmetric bilinear form on g is a scalar multiple of the Killing form.


  1. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function.
  2. The conformally invariant scalar curvature is the trace of Ric n m; i.e., the function where f is given by Eq.
  3. It involves the spectral theorem and the reason why it can be better to view a scalar as the trace of a matrix than as a mere scalar.


  1. The basic operations that can be performed with vectors are addition, subtraction, and scalar multiplication.
  2. These are: 1) the zero element, 2) addition, 3) subtraction, 4) scalar multiplication, and 5) the inner product.
  3. Behavior under addition, subtraction, scalar multiplication Suppose v 1 and v 2 are known pseudovectors, and v 3 is defined to be their sum, v 3 = v 1 + v 2.


  1. It follows directly from its definition that the scalar product is commutative.
  2. Scalar multiplication is both associative and commutative.


  1. Scalar (0th rank tensor) fields have spin 0.
  2. Since the Higgs field is a scalar field, the Higgs boson has no spin, hence no intrinsic angular momentum.
  3. For example, in Yang-Mills theories (such as quantum chromodynamics) the ghosts are complex scalar fields (spin 0), but they anti-commute (like fermions).


  2. In either view, as the scalar field slowly relaxes to the vacuum, the cosmological constant goes to zero, and space begins to expand normally.
  3. If g'=fg for some scalar function - the conformal change of metric - then W'=fW. For constant curvature, the Weyl tensor is zero.

Inner Product

  2. The dot product (or scalar) product is a typical example of an inner product.


  1. The existence of a scalar product does not imply the existence of a norm, so having an inner product is not a strictly stronger condition than having a norm.


  1. In supersymmetric theories the axion has both a scalar and a fermionic superpartner.
  2. The most famous solution that has been proposed to solve the strong CP problem is the Peccei-Quinn theory, involving new scalar particle s called axion s.
  3. The most famous solution that has been proposed to solve the strong CP problem is the Peccei-Quinn theory, involving new scalar particles called axion s.

Quantum Field Theory

  1. The name scalar boson arises from quantum field theory.
  2. Almost any textbook on quantum field theory (QFT) will contain a description of both classical and quantum scalar field theory.
  3. The philosophy here is the same as in starting a course on quantum field theory with a thorough study of scalar field theory.

Higgs Mechanism

  1. Another phenomenon commonly studied in the context of the scalar field is in the Higgs mechanism.
  2. Through the Higgs mechanism the Lagrangian is transformed into one real scalar, three massive vector and one massless vector boson.

Standard Model

  1. The Standard Model uses all of the available field types, scalar, vector and spinor as well as various gauge groups.
  2. Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model [ 4], [ 5].


  1. A Riemannian metric is a generalisation of the usual idea of the scalar or dot product to a manifold.
  2. If the manifold M has a metric of positive scalar curvature and b 2 +(M)≥2 then all Seiberg-Witten invariants of M vanish.


  1. A metric on a linear space defines an inner product which combines two vectors to produce a scalar.
  2. Metric (scalar or inner product) on a vector bundle.
  3. Scalar curvature The n -dimensional torus does not admit a metric with positive scalar curvature.


  1. Linear map - In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
  2. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity.
  3. In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.

Linear Algebra

  1. Scalar multiplication - In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra.


  1. We also introduced some basic concepts and notations: transpose, inverse, scalar (inner) product, outer product, and LU-factorization.


  2. The map π is not an isomorphism, since it maps any scalar multiple of to the same transformation.


  1. It is symmetric, scalar valued, and can be used to determine the angle between two vectors: in particular if a and b are orthogonal it is zero.
  2. The set of all vectors such that, s denoting a given (real) scalar, determines a plane P orthogonal to.
  3. The gradients of these scalar functions are orthogonal and have interpretation in terms of local polarization direction and momentum direction.


  1. In linear algebra, a Euclidean subspace (or subspace of R n) is a set of vectors that is closed under addition and scalar multiplication.


  1. But, the eigenvalues of A here are all 1, and only has one eigenvector, [1, 0, 0] and its scalar multiples.
  2. If called in scalar context it hands back only the eigenvalues.


  1. If an operator or function expects an operand to be a scalar, we say that the operand or argument is being evaluated in a scalar context.
  2. In vector calculus, divergence is an operator that measures the magnitude of a vector field 's source or sink at a given point, in terms of a signed scalar.
  3. This operator often makes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric.


  1. An eigenvalue of a n by n matrix A is a scalar c such that A* x = c* x holds for some nonzero vector x (where x is an n -tuple).
  2. The scalar λ is said to be an eigenvalue of A corresponding to the eigenvector x.
  3. Notice that a scalar is an eigenvalue of if and only if the linear transformations has a nontrivial kernel.


  1. The quadratic form on a scalar is just Q(λ) = λ2.


  1. If every element in a linear space has the defined norm which is scalar in the space, then the space is said to be "normed linear space".
  2. There are a large number of people who think a physical scalar is just a number and that a vector is a number along with a direction in physical space.


  1. Mathematics > Algebra > Linear Algebra > Vectors
  2. Multiplication
  4. Quantity
  5. Glossaries > Glossary of Linear Algebra /

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  Originally created: June 13, 2008.
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