Search results
Results from the WOW.Com Content Network
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.
On the other hand, this makes the Frobenius normal form rather different from other normal forms that do depend on factoring the characteristic polynomial, notably the diagonal form (if A is diagonalizable) or more generally the Jordan normal form (if the characteristic polynomial splits into linear factors). For instance, the Frobenius normal ...
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.
If A and B are unitarily equivalent, then tr AA* = tr BB*, where tr denotes the trace (in other words, the Frobenius norm is a unitary invariant). This follows from the cyclic invariance of the trace: if B = U *AU, then tr BB* = tr U *AUU *A*U = tr AUU *A*UU * = tr AA*, where the second equality is cyclic invariance. [3]
Download QR code; Print/export Download as PDF; Printable version; In other projects ... The Frobenius norm of A can be computed by the eigenvalues of A: ...
In mathematics, especially representation theory and combinatorics, a Frobenius characteristic map is an isometric isomorphism between the ring of characters of symmetric groups and the ring of symmetric functions. It builds a bridge between representation theory of the symmetric groups and algebraic combinatorics. This map makes it possible to ...
Download as PDF; Printable version; ... the Frobenius companion matrix of the monic polynomial () ... and the eigenvalues of largest complex norm give an asymptotic ...
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 ...