ORTHONORMALITY Meaning and
Definition
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Orthonormality is a mathematical concept used in linear algebra to describe a special type of relationship between vectors or functions. When a set of vectors or functions is orthonormal, it means that they are both orthogonal and have unit norm.
Orthogonality refers to the mathematical property of being perpendicular or independent of each other. In the context of vectors, two vectors are orthogonal if their dot product is zero, indicating that they are at a right angle to each other. Similarly, in the context of functions, two functions are orthogonal if their integral over a given interval is zero.
Unit norm refers to the property of having a magnitude or length equal to one. In the case of vectors, it can be determined by calculating the square root of the sum of the squares of its components. For functions, the unit norm is defined as the square root of the integral of the function's square over its domain.
When a set of vectors or functions is both orthogonal and each individual vector or function has unit norm, they are considered orthonormal. This means that they are not only mutually perpendicular or independent of each other but also have the same magnitude of one. Orthonormality is a desired property as it simplifies various mathematical calculations and manipulations, particularly in solving systems of linear equations, working with Fourier transforms, constructing orthogonal projections, and more.
Etymology of ORTHONORMALITY
The word "orthonormality" is a combination of two terms: "ortho-" and "normality".
The prefix "ortho-" is derived from the Greek word "orthos" meaning "straight" or "right". In mathematics, this prefix is often used to indicate perpendicularity or orthogonality.
The term "normality" comes from the adjective "normal" which originated from the Latin word "normalis", meaning "made according to a carpenter's rule" or "right-angled". Over time, "normal" evolved to refer to something that conforms to a standard or is perpendicular to a given surface or line.
When these two terms are combined, "orthonormality" refers to a mathematical concept in linear algebra. Orthonormality describes the geometric property of a set of vectors or functions that are both orthogonal (perpendicular to one another) and normalized (having a unit length or magnitude).
