Prob 5 - ans 3 good

Prob 5 - ans 3 good - Math H110 NORMlite 5:50 pm Notes on...

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Math. H110 NORMlite November 14, 1999 5:50 pm Prof. W. Kahan Page 1/21 Notes on Vector and Matrix Norms These notes survey most important properties of norms for vectors and for linear maps from one vector space to another, and of maps norms induce between a vector space and its dual space. Dual Spaces and Transposes of Vectors Along with any space of real vectors x comes its dual space of linear functionals w T . The representation of x by a column vector x , determined by a coordinate system or Basis , is accompanied by a corresponding way to represent functionals w T by row vectors w T so that always w T x = w T x . A change of coordinate system will change the representations of x and w T from x and w T to x = C –1 x and w T = w T C for some suitable nonsingular matrix C , keeping w T x = w T x . But between vectors x and functionals w T no relationship analogous to the relationship between a column x and the row x T that is its transpose necessarily exists. Relationships can be invented; so can any arbitrary maps between one vector space and another. For example, given a coordinate system, we can define a functional x T for every vector x by choosing arbitrarily a nonsingular matrix T and letting x T be the functional represented by the row (Tx) T in the given coordinate system. This defines a linear map x T = T ( x ) from the space of vectors x to its dual space; but whatever change of coordinates replaces column vector x by x = C –1 x must replace (Tx) T by ( Tx ) T = (Tx) T C = (TC x ) T C to get the same functional x T . The last equations can hold for all x only if T = C T TC . In other words, the linear map T ( x ) defined by the matrix T in one coordinate system must be defined by T = C T TC in the other. This relationship between T and T is called Congruence ( Sylvester’s word for it ). Evidently matrices congruent to the same matrix are congruent to each other; can all matrices congruent to a given matrix T be recognized? Only if T = T T is real and symmetric does this question have a simple answer; it is Sylvester’s Law of Inertia treated elsewhere in this course. The usual notation for complex vector spaces differs slightly from the notation for real spaces. Linear functionals are written w H or w * instead of w T , and row vectors are written w H or w* to denote the complex conjugate transpose of column w instead of merely its transpose w T . ( Matlab uses w.’ for w T and w’ for w* .) We’ll use the w* notation because it is older and more widespread than w H . Matrix T is congruent to C*TC whenever C is any invertible matrix and C* is its complex conjugate transpose. Most theorems are the same for complex as for real spaces; for instance Sylvester’s Law of Inertia holds for congruences among complex Hermitian matrices T = T* as well as real symmetric. Because many proofs are simpler for real spaces we shall stay mostly with them.
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Prob 5 - ans 3 good - Math H110 NORMlite 5:50 pm Notes on...

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