Scientific Computing

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CS205 Review Session #2 Notes Symmetric matrices and dot product Lemma An n × n matrix A is symmetric ⇐⇒ ∀ x , y IR n x · Ay = y · Ax Proof In the forward direction, if A is symmetric we have: x · Ay = x T Ay = ( x T Ay ) T = y T A T x = y T Ax = y · Ax For the converse, recall that e i is the i -th cartesian basis vector. Taking x = e i and y = e j and then the reverse, we have: e i · A e j = e T i Ae j = A ij e j · A e i = e T j Ae i = A ji Thus A ij = A ji for all i, j and A is symmetric. Lemma If A , B are symmetric n × n matrices and x IR n x T Ax = x T Bx then A = B Proof First, take x = e i : e T i Ae i = e T i Be i A ii = B ii Similarly, taking x = e i + e j gives: ( e i + e j ) T A ( e i + e j ) = ( e i + e j ) T B ( e i + e j ) A ii + A jj + A ij + A ji = B ii + B jj + B ij + B ji 2 A ij = 2 B ij Fundamental Subspaces Recall that an n × n matrix A is just a basis for a vector space. If the columns of A are all linearly independent, then A is a basis for IR n . If some columns are linear combinations of others, then the maximal subset of columns that are linearly independent forms a basis for a subspace. The subspace spanned by the columns is called the column space of the matrix, and is given as col( A ) = { Ax | x IR n } . The rank of A is the number of linearly
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