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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.
The matrix of the linear operator with respect to such a basis is the companion matrix of a monic polynomial; this polynomial (the minimal polynomial of the operator restricted to the subspace, which notion is analogous to that of the order of a cyclic subgroup) determines the action of the operator on the cyclic subspace up to isomorphism, and ...
A Frobenius matrix is a special kind of square matrix from numerical analysis. A matrix is a Frobenius matrix if it has the following three properties: all entries on the main diagonal are ones; the entries below the main diagonal of at most one column are arbitrary; every other entry is zero; The following matrix is an example.
The norm derived from this inner product is called the Frobenius norm, and it satisfies a submultiplicative property, as can be proven with the Cauchy–Schwarz inequality: [ ()] , if A and B are real matrices such that A B is a square matrix. The Frobenius inner product and norm arise frequently in matrix calculus and statistics.
Matrix analysis; Matrix multiplication; Matrix norm; Tensor product of Hilbert spaces – the Frobenius inner product is the special case where the vector spaces are finite-dimensional real or complex vector spaces with the usual Euclidean inner product
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 nilpotent part N is generally not unique either, but its Frobenius norm is uniquely determined by A (just because the Frobenius norm of A is equal to the Frobenius norm of U = D + N). [6] It is clear that if A is a normal matrix, then U from its Schur decomposition must be a diagonal matrix and the column vectors of Q are the eigenvectors of A.
where ‖ ‖ denotes the Frobenius norm. This is a special case of Wahba's problem (with identical weights; instead of considering two matrices, in Wahba's problem the columns of the matrices are considered as individual vectors). Another difference is that Wahba's problem tries to find a proper rotation matrix instead of just an orthogonal one.