2.30. For p1 and ox, prove that the matrix p-norms satisfy the five properties given near the end of Section 2.3.2 2.3.2 Matrix Norms We also need some way to measure the size or magnitude of matrices. Again, a more general definition is possible, but all of the matrix norms we will use are 2.3 Sensitivity and Conditioning defined in terms of an underlying vector norm. Specifically, given a vector norm, we define the corresponding matrix norm of an m x n matrix A by ||A||= max 20 ||Ax|| Such a matrix norm is said to be induced by or subordinate to the vector norm. Intuitively, the norm of a matrix measures the maximum stretching the matrix does to any vector, as measured in the given vector norm. Some matrix norms are much easier to compute than others. For example, the matrix norm corresponding to the vector 1-norm is simply the maximum absolute column sum of the matrix, m ||A||1=maxla i=1 and the matrix norm corresponding to the vector co-norm is simply the maximum absolute row sum of the matrix, ||A||∞ = maxΣlaijl. j=1 A handy way to remember these is that the matrix norms agree with the cor- responding vector norms for an n x 1 matrix. Unfortunately, the matrix norm corresponding to the vector 2-norm is not so easy to compute (see Section 3.6.1).