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Solutions for Chapter 8.16: Method of Least Squares

Full solutions for Advanced Engineering Mathematics | 6th Edition

ISBN: 9781284105902

Solutions for Chapter 8.16: Method of Least Squares

Solutions for Chapter 8.16
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ISBN: 9781284105902

Chapter 8.16: Method of Least Squares includes 10 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by Patricia and is associated to the ISBN: 9781284105902. Since 10 problems in chapter 8.16: Method of Least Squares have been answered, more than 12240 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6.

Key Math Terms and definitions covered in this textbook
• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Dimension of vector space

dim(V) = number of vectors in any basis for V.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• Hilbert matrix hilb(n).

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• Linear combination cv + d w or L C jV j.

• Lucas numbers

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Outer product uv T

= column times row = rank one matrix.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

• Semidefinite matrix A.

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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