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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.
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 2-norm is the Frobenius norm. ... The latter version of Hölder's inequality is proven in higher ... Theory of Quantum Information, 2.3 Norms of ...
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]
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.
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.
Frobenius reciprocity theorem in group representation theory describing the reciprocity relation between restricted and induced representations on a subgroup Perron–Frobenius theorem in matrix theory concerning the eigenvalues and eigenvectors of a matrix with positive real coefficients