By Jorge Rebaza

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Example text
In addition, their condition number is I, the best possible. Now we introduce one of the most important orthogonal matrices that are extensively used in matrix computations and that we will use explicitly in Chapter 3. This matrix has the additional and convenient property of symmetry. 23 (Householder matrix). Let ubea unit vector in R n . 29) is called Householder matrix. 29) is orthogonal and symmetric. Proof. Since (uuT)T = uuT, we have HT = H and hence H is symmetric. In addition, since uTu = 1, we get HTH = {I 2uuT)T{I  2uuT) = ( I  2uuT) (I  2uuT) = I — AuuT + 4u uTu uT — J, and therefore H is orthogonal.
14 The matrix ATA is symmetric positive semidefinite, for any matrix Amxn. 15 a b The matrix A = b a with a > 0 is positive definite. 17 Let A be a square matrix of order n, and let m < n. A principal submatrix of A is a matrix obtained by deleting any n — m rows and corresponding columns. A leading principal submatrix is obtained by deleting the last n — m rows and columns. Notation. We can use and index set / C { 1 , . . , n} to represent which rows and columns of A are used to form the principal submatrix.
To show linear independence, assume that c\V\ + C2^2 = [0 0] T . Then, this equality gives the system 2ci + c2 c\ + 3c2 = = 0 0, whose solution is ci = c2 = 0. Therefore, vi and V2 are linearly independent. Observe that in this case, linear independence means the vectors are not parallel or. multiple of each other. 4. 36 is not linearly independent. In fact, any of the vectors in Si can be written as a combination of the other two. 37 is linearly independent. 41 Let V = P2 be the vector space of real polynomials of degree at most 2.