ORTHOGONAL GROUP Meaning and
Definition
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The orthogonal group is a mathematical term that refers to a specific group of linear transformations. Specifically, it is the set of all invertible linear transformations that preserve the dot product of vectors. In other words, it consists of all square matrices with real or complex entries that satisfy a certain condition.
More formally, the orthogonal group is denoted as O(n, F), where n represents the dimension of the vector space and F represents the field over which the vector space is defined (usually either the real numbers or the complex numbers). The orthogonal group O(n, F) consists of all n x n matrices A that satisfy the equation A^T * A = I, where A^T represents the transpose of A and I represents the identity matrix.
Geometrically, elements of the orthogonal group can be thought of as transformations that preserve distances and angles between vectors. This means that if a matrix A belongs to the orthogonal group, then for any two vectors u and v, the dot product of A * u and A * v will be equal to the dot product of u and v. Essentially, the orthogonal group captures the idea of preserving geometric properties of vectors under certain transformations.
The orthogonal group has numerous applications in various fields of mathematics and physics, including linear algebra, quantum mechanics, and signal processing. It plays a crucial role in the study of symmetry, rotations, and transformations in high-dimensional spaces.
Etymology of ORTHOGONAL GROUP
The term "orthogonal" in the context of the "orthogonal group" comes from the Greek words "ortho" meaning "straight" or "right", and "gonia" meaning "angle".
The concept of orthogonality is related to perpendicularity in geometry, where two lines are said to be orthogonal if they intersect at right angles. In linear algebra, the orthogonal group refers to a group of matrices that preserve the notion of orthogonality.
Therefore, the term "orthogonal group" is used to describe a group of matrices that preserve the notion of right angles or orthogonality.
