Advanced Engineering Mathematics 6th Edition - Solutions by Chapter

Parentheses can be removed to leave ABC.

The n roots are the eigenvalues of A.

Triangular matrix with one extra nonzero adjacent diagonal.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

A sequence of steps intended to approach the desired solution.

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Every v in V is orthogonal to every w in W.

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

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 ].

Signs in A = signs in D.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

T- 1 has rank 1 above and below diagonal.

Orthonormal columns (complex analog of Q).

The rows (or the columns) of A generate a box with volume I det(A) I.