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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.
where [] is the augmented matrix with E and F side by side and ‖ ‖ is the Frobenius norm, the square root of the sum of the squares of all entries in a matrix and so equivalently the square root of the sum of squares of the lengths of the rows or columns of the matrix. This can be rewritten as
The solution with minimum Euclidean norm is . [27] This result is easily extended to systems with multiple right-hand sides, when the Euclidean norm is replaced by the Frobenius norm. Let B ∈ K m × p {\displaystyle B\in \mathbb {K} ^{m\times p}} .
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
Hadamard product (matrices) Hilbert–Schmidt inner product; Kronecker product; 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
Every inner product space induces a norm, called its canonical norm, that is defined by ‖ ‖ = , . With this norm, every inner product space becomes a normed vector space. So, every general property of normed vector spaces applies to inner product spaces.
The Frobenius inner product may be extended to a hermitian inner product on the complex vector space of all complex matrices of a fixed size, by replacing B by its complex conjugate. The symmetry of the Frobenius inner product may be phrased more directly as follows: the matrices in the trace of a product can be switched without changing the ...
The Frobenius norm is the Hilbert-Schmidt norm, but it is not the same as ‖ ‖ = (this is the 'spectral norm'). For vectors, ‖ a ‖ 2 {\displaystyle \|a\|_{2}} is the Euclidean norm which is the same as the Frobenius norm if the input vector is treated like a matrix, but when the input is a matrix, the notation ‖ A ‖ 2 {\displaystyle ...