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The Schatten p-norms arise when applying the p-norm to the vector of singular values of a matrix. [2] If the singular values of the m × n {\displaystyle m\times n} matrix A {\displaystyle A} are denoted by σ i , then the Schatten p -norm is defined by
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, ... denotes the Frobenius norm.
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 ...
For example, the Ky Fan-k-norm is the sum of first k singular values, the trace norm is the sum of all singular values, and the Schatten norm is the pth root of the sum of the pth powers of the singular values. Note that each norm is defined only on a special class of operators, hence singular values can be useful in classifying different ...
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 ginv function calculates a pseudoinverse using the singular value decomposition provided by the svd ... when the Euclidean norm is replaced by the Frobenius norm.
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.
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]