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However, normal may also refer to the magnitude of a vector. In particular, a set is called orthonormal (orthogonal plus normal) if it is an orthogonal set of unit vectors. As a result, use of the term normal to mean "orthogonal" is often avoided. The word "normal" also has a different meaning in probability and statistics.
The line segments AB and CD are orthogonal to each other. In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.Whereas perpendicular is typically followed by to when relating two lines to one another (e.g., "line A is perpendicular to line B"), [1] orthogonal is commonly used without to (e.g., "orthogonal lines A and B").
Suppose x is a Gaussian random variable with mean m and variance . Also suppose we observe a value y = x + w , {\displaystyle y=x+w,} where w is Gaussian noise which is independent of x and has mean 0 and variance σ w 2 . {\displaystyle \sigma _{w}^{2}.}
This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces.
Eigenvectors of a normal operator corresponding to different eigenvalues are orthogonal, and a normal operator stabilizes the orthogonal complement of each of its eigenspaces. [3] This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary operator.
In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized , the resulting basis is an orthonormal basis .
Conversely, if and are independent random variables and their sum + has a normal distribution, then both and must be normal deviates. [ 48 ] This result is known as Cramér's decomposition theorem , and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal.
The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1.