Numerical Analysis 10th Edition - Solutions by Chapter

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Columns without pivots; these are combinations of earlier columns.

Complex analog a j i = aU of a symmetric matrix.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

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.

= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Orthogonal Q times positive (semi)definite H.

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).

Appears in block elimination on [~ g ].

One free variable is Si = 1, other free variables = o.

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

Constant down each diagonal = time-invariant (shift-invariant) filter.

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.