What is "scalar"

Scalar \Sca"lar\, n. (Math.) In the quaternion analysis, a quantity that has magnitude, but not direction; -- distinguished from a vector, which has both magnitude and direction.

"resembling a ladder," 1650s, from Latin scalaris "of or pertaining to a ladder," from scalae (plural) "ladder, steps, flight of steps" (see scale (n.2)). Mathematical sense first recorded 1846.

a. 1 (context mathematics English) Having magnitude but not direction 2 (context computer science English) Consisting of a single value (e.g. integer or string) rather than multiple values (e.g. array) 3 Of, or relating to scale n. 1 (context mathematics English) A quantity that has magnitude but not direction; compare vector 2 (context electronics English) An amplifier whose output is a constant multiple of its input

adj. of or relating to a directionless magnitude; "scalar implicatures"

n. a variable quantity that cannot be resolved into components

Scalar may refer to:

• Scalar (mathematics), an element of a field, usually a real number, which is used to define a vector space
• Scalar (physics), a quantity represented by a mathematical scalar that is independent of specific classes of coordinate systems, or one that is usually said to be described by a single real number
• Lorentz scalar, a quantity in the theory of relativity which is invariant under a Lorentz transformation
• Variable (computing), or scalar, an atomic quantity that can hold only one value at a time

A scalar is an element of a field which is used to define a vector space. Scalars in physics are usually real numbers, or any quantity that can be measured using a single real number, such as temperature, length, and mass, and is usually said to have magnitude but no direction. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector.

In linear algebra, real numbers or other elements of a field are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. More generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers. Then the scalars of that vector space will be the elements of the associated field.

A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called an inner product space.

The real component of a quaternion is also called its scalar part.

The term is also sometimes used informally to mean a vector, matrix, tensor, or other usually "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1×n matrix and an n×1 matrix, which is formally a 1×1 matrix, is often said to be a scalar.

The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix.

A scalar in physics is a physical quantity that can be described by a single element of a number field such as a real number, often accompanied by units of measurement. A scalar is usually said to be a physical quantity that only has magnitude and no other characteristics. This is in contrast to vectors, tensors, etc. which are described by several numbers that characterize their magnitude, direction, and so on. A vector is usually said to be a physical quantity that has magnitude and direction. Formally, a scalar is unchanged by coordinate system rotations or reflections (in Newtonian mechanics), or by Lorentz transformations or space-time translations (in relativity). A related concept is a pseudoscalar, which is invariant under proper rotations but (like a pseudovector) flips sign under improper rotations. The concept of a scalar in physics is essentially the same as in mathematics. A physical scalar field is one type of more general fields, like vector fields, spinor fields, and tensor fields.

An example of a scalar quantity is temperature: the temperature at a given point is a single number. Velocity, on the other hand, is a vector quantity: velocity in three-dimensional space is specified by three values; in a Cartesian coordinate system the values are the speeds relative to each coordinate axis. The associated fields describe the temperature and velocity in each point of some space. Considering the norms of the velocity vectors results in a scalar field of the speeds in each point of the space.