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The Frobenius norm is an extension of the Euclidean norm to and comes from the Frobenius inner product on the space of all matrices. The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. The sub-multiplicativity of Frobenius norm can be proved using Cauchy–Schwarz inequality.
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models. The total least squares approximation of the data is generically equivalent to the best, in the Frobenius norm, low-rank approximation of the data matrix. [1]
One example is the squared Frobenius norm, which can be viewed as an -norm acting either entrywise, or on the singular values of the matrix: = ‖ ‖ = | | = =. In the multivariate case the effect of regularizing with the Frobenius norm is the same as the vector case; very complex models will have larger norms, and, thus, will be penalized ...
The Schatten 1-norm is the nuclear norm (also known as the trace norm, or the Ky Fan n-norm [1]). The Schatten 2-norm is the Frobenius norm. The Schatten ∞-norm is the spectral norm (also known as the operator norm, or the largest singular value).
The standard Golden–Thompson inequality is a special case of the above inequality, where the norm is the Frobenius norm. The general case is provable in the same way, since unitarily invariant norms also satisfy the Cauchy-Schwarz inequality. (Bhatia 1997, Exercise IV.2.7)
Human development theory is a theory which uses ideas from different origins, such as ecology, sustainable development, feminism and welfare economics. It wants to avoid normative politics and is focused on how social capital and instructional capital can be deployed to optimize the overall value of human capital in an economy.
In group theory, the induced representation is a representation of a group, G, which is constructed using a known representation of a subgroup H.Given a representation of H, the induced representation is, in a sense, the "most general" representation of G that extends the given one.