Errors - University of Michigan Dearborn
2.5 General Matrix Norms.
In section 2.3 we looked at the infinity and one norms and saw how they could be used to estimate the error in the solution of a system of linear equations y = Ax in terms of the errors in the numbers in y. In this section we look at general matrix norms where one may use some other norm for x and y. They can be used in the same way as the infinity and one norm to estimate the error in the solution of a system of linear equations.
Definition 1. Let A be a matrix and consider the mapping y = Ax. Choose a norm || x || for vectors x and choose a norm || Ax || for vectors Ax. Then the norm of A, denoted by || A ||, is the maximum value of the ratio as x varies over all non-zero vectors x, i.e.
(1) || A || =
It follows from (1) that
(2) || Ax || ( || A || || x ||
which is a generalization of (9) and (10) in section 2.3. In particular, || A || is the maximum stretching that A does when applied to vectors x.
Just as when we are dealing with the sup norm and one norm for x and y, (2) turns out to be useful when we want to analyze how errors propagate when we multiply by A. The following proposition generalizes Proposition 2 of section 2.3.
Proposition 1. Let A = and y = Ax and ya = Axa. Then
(3) || y - ya || ( || A || || x - xa ||
If m = n and A is invertible then
(4) || x - xa || ( || A-1 || || y - ya ||
Proof. (3) follows from (2) and the fact that y – ya = A(x – xa). (4) follows from (3) and the fact that x = A-1y and xa = A-1ya. //
Unfortunately, the formula (1) is not so nice for calculating the norm of some matrix A. Sometimes it is convenient to restrict the vectors x that we are maximizing over in (1) to unit vectors.
Proposition 2. The norm of A is the maximum of || Au || as u varies over unit vectors, i.e.
(5) || A || = || Au ||
Proof. Note that in (1) we can write
= || Ax || = || Ax || = || A || = || Au ||
where u = and we have used the homogeniety of the norm. Note that u is a unit vector, since
|| || = || x || = 1
So, when we maximize over all non-zero vectors x in (1), it is the same as maximizing over all unit vectors in (5). //
Here are some properties of the matrix norms.
Proposition 3. Let || A || be a matrix norm defined by (1). If A and B are matrices and c is a number then
(6) || A + B || ( || A || + || B ||
(7) || cA || = | c | || A ||
(8) || A || = 0 ( A = 0
(9) || AB || ( || A || || B ||
(10) || I || = 1
(11) || A-1 || (
Proof. To prove (6) we use (5).
|| A + B || = || (A + B)x || = || Ax + Bx || ( (|| Ax || + || Bx ||)
( || Ax || + || Bx || = || A || + || B ||
For (7) one has
|| cA || = || (cA)x || = || c(Ax) || ( | c | || Ax ||
( | c | || Ax || = | c | || A ||
To prove (8) note that || A || = 0 and (1) imply = 0 which implies || Ax || = 0 for all x which implies A = 0. Note that (1) implies || (AB)x || = || A(Bx) || ( || A || || Bx || ( || A || || B || || x ||. Therefore
( || A || || B || which implies ( || A || || B || which implies (9). (10) follows from (1) and the fact that Ix = x for all x. For (11) note that I = AA-1. Using (9) and (10) we get 1 = || I || ( || A || || A-1 || from which (11) follows. //
Even the formula (5) is not convenient for finding the norm of a matrix. For given norms || x || and || Ax || it may take some work to find a formula for the corresponding matrix norm. For the Euclidean norm here are two useful formulas.
Proposition 2. Let || x || and || Ax || be the Euclidean norms and || A || be associated matrix norm given by (1). Then
(6) || A || = max{ | ( |: ( is an eigenvalue of A} if A is symmetric
(7) || A || = max{ : ( is an eigenvalue of ATA} in general
Proof. First consider the case where A is symmetric. Then the eigenvalues (1, …, (n of A are all real and there is an orthonormal basis v1, …, vn of eigenvalues of A. Suppose Avi = (ivi and | (1 | ( … ( | (n |. For a given vector x one can write x = (1v1 + … + (nvn. Since the vi are orthonormal one has || x ||2 = One has Ax =
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