Definition
In what follows, will denote the field of real or complex numbers. Let denote the vector space containing all matrices with rows and columns with entries in . Throughout the article denotes the conjugate transpose of matrix .
A matrix norm is a vector norm on . That is, if denotes the norm of the matrix, then,
- if and iff
- for all in and all matrices in
- for all matrices and in
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:
- for all matrices and in
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-by-n matrices, together with such a sub-multiplicative norm, is an example of a Banach algebra.
Read more about this topic: Matrix Norm
Famous quotes containing the word definition:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)