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Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
Dual norm – Measurement on a normed vector space; Matrix norm – Norm on a vector space of matrices; Norm (mathematics) – Length in a vector space; Normed space – Vector space on which a distance is defined; Operator algebra – Branch of functional analysis
Using the pseudoinverse and a matrix norm, one can define a condition number for any matrix: = ‖ ‖ ‖ + ‖. A large condition number implies that the problem of finding least-squares solutions to the corresponding system of linear equations is ill-conditioned in the sense that small errors in the entries of A {\displaystyle A} can ...
Vector norm, a map that assigns a length or size to any vector in a vector space; Matrix norm, a map that assigns a length or size to a matrix; Operator norm, a map that assigns a length or size to any operator in a function space; Norm (abelian group), a map that assigns a length or size to any element of an abelian group
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
Notice that ‖ ‖ is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), ‖ ‖ is the trace class norm (see trace class), and ‖ ‖ is the operator norm (see operator norm). Note that the matrix p-norm is often also written as ‖ ‖, but it is not the same as Schatten norm.
To reflect a point through a plane + + = (which goes through the origin), one can use =, where is the 3×3 identity matrix and is the three-dimensional unit vector for the vector normal of the plane. If the L 2 norm of , , and is unity, the transformation matrix can be expressed as: = []