- Chapter 3.1: del(A + 8) = del(A) + del(B)
- Chapter 3.2: del(A - IB) = ~
- Chapter 3.3: If del(A) = O. then A has alleasl lwoequal rows.
- Chapter 3.4: If A a~ a column of all zeroS. then del(A} = O
- Chapter 3.5: A is singular if and only if det(A) = O
- Chapter 3.6: If 8 is the reduced row echelon fonn of A. then det(B) = del(A )
- Chapter 3.7: The determinant oran elementary millrix is always I.
- Chapter 3.8: If A is I1Ollsingular. then A - I _ ~ ~IJj(
- Chapter 3.9: If T i~ a matrix transformation frolll R.2 ..... R. 2 defined by A ...
- Chapter 3.10: If aJllhe diagonal elements of an /I X /I m.1lrix A are zero. then ...
- Chapter 3.11: det(ABTA - J)= detB.
- Chapter 3.12: - (del cA) = det(A).
Solutions for Chapter Chapter 3: Compute IAI for each of the followin g:
Full solutions for Elementary Linear Algebra with Applications | 9th Edition
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Every v in V is orthogonal to every w in W.
Orthonormal vectors q 1 , ... , q n·
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 •
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Singular Value Decomposition
(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.
Solvable system Ax = b.
The right side b is in the column space of A.
Constant down each diagonal = time-invariant (shift-invariant) filter.