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Using this fact, we now have | | g ^ | | 2 = max ||QAc|| 2 = max ||Ar|| 2 = \\A\\2. \\x\\2=i IFI|2=I □ The next important property of an orthogonal matrix is especially related to what is known as sensitivity in the solution of several problems. First we need the following definition. 21 The condition number of a square matrix A is defined as cond(A) = | | i 4 | | p - 1 | | . 28) This number represents how well or ill-posed a matrix can be, in the sense that how much we can rely on the computations performed with such a matrix, such as solving systems of equations, due to potential accumulation of round-off errors.

54) Furthermore, this approach is in fact true for a projection matrix onto any vector subspace S. 54), where Q is a matrix whose columns form an orthonormal basis of S. 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.

By instance, 8 3 4 1 0 2 -6 -1 1 8]T,v3 = [0 2 0] T } can be written as a unique combination -2 1 8 +2 0 2 0 This means that on the basis B above, the vector [8 3 4] T is fully represented by its coordinates { 6 , - 1 , 2}. 42, the same vector is represented by its corresponding coordinates {8, 3, 4}. 42, it is known as the canonical or standard basis of such space. This basis is particularly important (among other things) in image compression, which we study in Chapter 5. 45 The set B — { f\ — 1, $2 — x, f% — x2 } is a basis for the vector space P2 of the polynomials of degree at most two.