norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication. Given a field  ...

{R} ^{m},} then the matrix norm given to a matrix A {\displaystyle A} is the square root of the largest eigenvalue of the matrix A ∗ A {\displaystyle...

identity matrix, this norm corresponds to the Euclidean norm. If A {\displaystyle A} is diagonal, this norm is also called a weighted norm. The energy norm is...

infimum of all norms of a matrix. Indeed, on the one hand, ρ ( A ) ⩽ ‖ A ‖ {\displaystyle \rho (A)\leqslant \|A\|} for every natural matrix norm ‖ ⋅ ‖ {\displaystyle...

addition, the Frobenius norm and the trace norm (the nuclear norm) are special cases of the Schatten norm. The singular values of a matrix ⁠ A {\displaystyle...

choice of norm, as can be illustrated by two examples. If ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the matrix norm induced by the (vector) Euclidean norm (sometimes...

the vector norm enforcing a regularization penalty on x {\displaystyle x} has been extended to a matrix norm on X {\displaystyle X} . Matrix regularization...

In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal...

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express...

\left(A_{n}\right)} ⁠ converges to the matrix ⁠ A {\displaystyle A} ⁠ (in the maximum norm or Frobenius norm, say), then ⁠ ( A n ) + {\displaystyle (A_{n})^{+}}...

matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes...

then the corresponding matrix series converges for matrices A such that ‖ A ‖ < r {\displaystyle \|A\|<r} for some matrix norm that satisfies ‖ A B ‖...

the matrix measure or the Lozinskiĭ measure. Let A {\displaystyle A} be a square matrix and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} be an induced matrix norm. The...

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...

the same as Schatten norm. In fact, we have ‖ A ‖ matrix p-norm , 2 = ‖ A ‖ Schatten , ∞ {\displaystyle \|A\|_{{\text{matrix p-norm}},2}=\|A\|_{{\text{Schatten}}...

Lax equivalence theorem. Stability in this context means that a matrix norm of the matrix used in the iteration is at most unity, called (practical) Lax–Richtmyer...

rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [...

normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved...

real matrices such that A B is a square matrix. The Frobenius inner product and norm arise frequently in matrix calculus and statistics. The Frobenius...

particular matrix. The norm of a matrix is a positive real number. For all matrices A and B in Mmn(F), and all numbers α in F, a matrix norm, delimited...

{\displaystyle CD} " norm: ‖ A ‖ {\displaystyle \|A\|} , read "the norm (length, size, magnitude etc.) of the matrix A {\displaystyle A} ". The norm of a one-dimensional...

{\displaystyle n} goes to infinity, the matrix norm of C {\displaystyle C} must be smaller than unity. This norm is | | C | | ∞ = max i ∑ j | c i j | =...

abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set...

{\displaystyle P} and I − P {\displaystyle I-P} are equal, and thus that the matrix norm of the oblique projections are the same. However, the condition number...

subset of n-by-n matrix units in the set of n-by-n matrices over R. The matrix norm (induced by the same two vector norms) of a matrix unit is equal to...

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Let K be a field...

example, in the low-rank matrix completion problem one may apply the regularization penalty taking the form of a nuclear norm R ( X ) = λ ‖ X ‖ ∗ {\displaystyle...

greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix that may also be referred to as Smith's matrix. The study was initiated...

vector n-norm or matrix norm. Common examples of such norms include the L1 norm (sum of absolute component values), the L2 norm (Euclidean norm, or square...

linear algebra, a square matrix A {\displaystyle A}  is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists...