By Jorge Rebaza
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Using the adjoint method, we have A'1 = adjoint (A) /det (A). However, this method is rarely used in practice, and we do not discuss it here. 1). In general, the computation of the inverse of a matrix has to be done numerically, and great care has to be taken due to potentially large accumulation of errors for some matrices. Thus, in practice it is customary to avoid computing the inverse of a matrix explicitly, and some other options must be used. We will discuss these issues later on. 4 Symmetric and positive definite matrices A square matrix A of order n is called symmetric if A = A7', that is, the matrix equals its transpose.
Later we will see that for a general m x n matrix A, with m > n, the least squares solution to the system Ax = b is given by the solution of the so-called normal equations ATAx = ATb. This is nothing else but a consequence of projecting b orthogonally onto col(A), as it will be explained in detail in Chapter 4. 54) x = (ATA)~1ATb. 52) is called the pseudoinverse matrix of A. 53) has taken the vector b onto co\(A), so we suspect such a matrix is an orthogonal projection ontocol(A). In fact, P2 = A(ATA)-1ATA(ATA)-1AT = A(ATA)-1AT = P, and since AT A is symmetric, PT = A(ATA)-TAT = A(ATA)-1AT = P.
62. 66 Let S be the subspace spanned by v\ = [1 2 3] T , and v2 = [1 1 1] T . 40) to obtain the vectors qi = [1 2 3] T /VTi, q2 = [4 1 - 2] T /V21. Then, we define the matrix Q with q\ and q2 as first column and second column respectively, so that the projection matrix onto S is given by I 5 2 - 1 2 -1 where Q = [qi subspace S. 2 2 2 5 q2]. 65. The vectors vi = [1 0 0 0] T , v2 = [0 1 0 1] T , v3 = [0 0 1 0] T form a basis of co\(A). They can be orthonormalized using Gram-Schmidt to get Ql = [1 0 0 0] T , q2 = [0 Now define the matrix Q = [qi q2 T P = QQ 1/V2 0 l / \ / 2 ] T , 93 = [0 0 1 0] T .