Trigonometry 11th Edition - Solutions by Chapter

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

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

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).

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Ax = AX with x#-O so det(A - AI) = o.

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

Use AT for complex A.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Diagonal entries = 1, off-diagonal entries = 0.

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 .

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

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 •

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

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

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Column vectors by convention.

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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