Prove that the spectral radius () is not a matrix norm on Mn by giving

examples of it failing to satisfy three of the four properties required of a matrix norm (positivity, homogeneity, subadditivity, sub-multiplicativity), and prove that it satisfies one of these four properties

A function III . [| : M. - R is a matrix norm if, for all A, B E M,, it satisfies the following five axioms: (1) |All 20 Nonnegative (la) |AllI = 0 if and only if A = 0 Positive (2) IcAlll = Ic| ||Alll for all ce C Homogeneous 5.6 Matrix norms 341 (3) HA + BIll EmAll + IBIII Triangle Inequality (4) HABILIS WAI IB III Submultiplicativity A matrix norm is sometimes called a ring norm. The first four properties of a matrix norm are identical to the axioms for a norm (5.1.1). A norm on matrices that does not satisfy property (4) for all A and B is a vector norm on matrices, sometimes called a generalized matrix norm. The notions of a matrix seminorm and a generalized matrix seminorm may also be defined via omission of axiom (la). Since IIIA3III = WIAAIII SHIAIII IIIAIII = | All for any matrix norm, it follows that | | All| > 1 for any nonzero matrix A for which AZ = A. In particular, I|/ l| 2 1 for any matrix norm. If A is nonsingular, then / = AA-, so 1/|/Il| = IIIAA-Ill s III Alll III All, and we have the lower bound 11|A'IIZDefinition 5.1.1. Let V be a vector space over the field F (F = R or C). A function 11 . 1 : V - R is a norm (sometimes one says vector norm) if, for all x, y e V and all CEF, (1) 1x1/20 Nonnegativity (la) |x) = 0 if and only if x = 0 Positivity (2) lexll = llllxll Homogeneity (3) |x+ yl = Ixl + yl Triangle Inequality These four axioms express some of the familiar properties of Euclidean length in the plane. Euclidean length possesses additional properties that cannot be deduced from these four axioms; an example is the parallelogram identity (5.1.9). The triangle inequality expresses the subadditivety of a norm. If || . || is a norm on a real or complex vector space V , the positivity and homogeneity axioms (la) and (2) ensure that any nonzero vector x can be normalized to produce a unit vector u = (xx: lal = Illx xll = x x/| = 1. A real or complex vector space V, together with a given norm | . II, is called a normed linear space (normed vector space). A function || - || : V - R that satisfies axioms (1). (2), and (3) of (5.1.1) is called a seminorm. The seminorm of a nonzero vector can be zero