Matrix norm

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). ‖ A r ‖ 1 / r ≥ ρ ( A ) , {displaystyle |A^{r}|^{1/r}geq ho (A),}     (1) In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). In what follows, K {displaystyle K} will denote a field of either real or complex numbers. Let K m × n {displaystyle K^{m imes n}} denote the vector space of all matrices of size m × n {displaystyle m imes n} (with m {displaystyle m} rows and n {displaystyle n} columns) with entries in the field K {displaystyle K} . A matrix norm is a norm on the vector space K m × n {displaystyle K^{m imes n}} . Thus, the matrix norm is a function ‖ ⋅ ‖ : K m × n → R {displaystyle |cdot |:K^{m imes n} o mathbb {R} } that must satisfy the following properties: For all scalars α {displaystyle alpha } in K {displaystyle K} and for all matrices A {displaystyle A} and B {displaystyle B} in K m × n {displaystyle K^{m imes n}} , Additionally, in the case of square matrices (thus, m = n), some (but not all) matrix norms satisfy the following condition, which is related to the fact that matrices are more than just vectors: A matrix norm that satisfies this additional property is called a sub-multiplicative norm (in some books, the terminology matrix norm is used only for those norms which are sub-multiplicative). The set of all n × n {displaystyle n imes n} matrices, together with such a sub-multiplicative norm, is an example of a Banach algebra. The definition of sub-multiplicativity is sometimes extended to non-square matrices, for instance in the case of the induced p-norm, where for A ∈ K m × n {displaystyle Ain {K}^{m imes n}} and B ∈ K n × k {displaystyle Bin {K}^{n imes k}} holds that ‖ A B ‖ q ≤ ‖ A ‖ p ‖ B ‖ q {displaystyle |AB|_{q}leq |A|_{p}|B|_{q}} . Here ‖ ⋅ ‖ p {displaystyle |cdot |_{p}} and ‖ ⋅ ‖ q {displaystyle |cdot |_{q}} are the norms induced from K n {displaystyle K^{n}} and K k {displaystyle K^{k}} , respectively, and p,q ≥ 1.