Numerical Analysis 10th Edition - Solutions by Chapter

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

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.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Each equation gives a plane in Rn; the planes intersect at x.

Column vectors by convention.

Appears in block elimination on [~ g ].

Every B = M-I AM has the same eigenvalues as A.

(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Columns of n by n identity matrix (written i ,j ,k in R3).

Any vector space inside V, including V and Z = {zero vector only}.

Signs in A = signs in D.

Orthonormal columns (complex analog of Q).