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The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
The Frobenius norm defined by ‖ ‖ = = = | | = = = {,} is self-dual, i.e., its dual norm is ‖ ‖ ′ = ‖ ‖.. The spectral norm, a special case of the induced norm when =, is defined by the maximum singular values of a matrix, that is, ‖ ‖ = (), has the nuclear norm as its dual norm, which is defined by ‖ ‖ ′ = (), for any matrix where () denote the singular values ...
Let Ω ⊂ R n be open and bounded, and let k ∈ N.Suppose that the Sobolev space H k (Ω) is compactly embedded in H k−1 (Ω). Then the following two norms on H k (Ω) are equivalent:
Crouzeix's conjecture is an unsolved problem in matrix analysis.It was proposed by Michel Crouzeix in 2004, [1] and it can be stated as follows: ‖ ‖ | |, where the set () is the field of values of a n×n (i.e. square) complex matrix and is a complex function that is analytic in the interior of () and continuous up to the boundary of ().
Pages for logged out editors learn more. Contributions; Talk; 2-norm
In statistics, the Bhattacharyya distance is a quantity which represents a notion of similarity between two probability distributions. [1] It is closely related to the Bhattacharyya coefficient, which is a measure of the amount of overlap between two statistical samples or populations.
In statistics, k-medians clustering [1] [2] is a cluster analysis algorithm. It is a generalization of the geometric median or 1-median algorithm, defined for a single cluster. k -medians is a variation of k -means clustering where instead of calculating the mean for each cluster to determine its centroid , one instead calculates the median .
The norm (see also Norms) can be used to approximate the optimal norm via convex relaxation. It can be shown that the L 1 {\displaystyle L_{1}} norm induces sparsity. In the case of least squares, this problem is known as LASSO in statistics and basis pursuit in signal processing.