- 2.2.1: If A b a Lc~lic matrix and ,. is a population di,tribution. then ea...
- 2.2.2: For any population disuibution v, if tl is the Leslie matrix for th...
- 2.2.3: In a Lc~lie matrix. the (i .j)entry equals the average num ber of f...
- 2.2.4: n a l..c>lie matrix. the (i + I. i)-entry cquab the portion of fema...
- 2.2.5: The applicalion in I his sec1ion on traffic flow relies on the asso...
- 2.2.6: If A and 8 are matrices and x, y. and z are vectors such I. If A b ...
- 2.2.7: A (0. 1)-malrix is a matrix with Os and Is as its only e nlries
- 2.2.8: A (0, I )-matrix is a square matrix with Os and Is as its only entries
- 2.2.9: Every (0, I )matrix is a symmetric malrix.
- 2.2.10: By observing a certain colony of mice. reearchcrs found that all an...
- 2.2.11: Suppose that the females of a certain colony of animals arc divided...
- 2.2.12: A certain colony of lizards has a life span of less than 3 years. S...
- 2.2.13: A certain colony of bat> has a life span of less than 3 years. Supp...
- 2.2.14: A certain colony of voles has a life span of less than 3 years. Sup...
- 2.2.15: A cenain colony of squirrels has a life span of less than 3 years. ...
- 2.2.16: The maximum membership term for each member of the Service Club is ...
- 2.2.17: A cenain medical foundation rccetves money from two sources: donatt...
- 2.2.18: Water is pumped into a system of pipes at points P1 and Pz shown in...
- 2.2.19: With the interpretation of a (0. I )-matrix found in Prnctice 3. ~u...
- 2.2.20: Recall the (0. 1 )-matrix ~ [l 0 l1 0 0 l 0 0 0 0 0 I 0 in which th...
- 2.2.21: Suppose that there is a group of four people and an associated -' x...
- 2.2.22: Suppo-c that student preference for a ;.et of courses is given in t...
- 2.2.23: Let A be the matrix in E'ample 2. (a) Justify the following interpr...
- 2.2.24: Sup1X>se that A and 8 arc m x m matrices that commute: that is. AB ...
- 2.2.25: In reference to the apphcallon in the text involving the N:1tchel I...
- 2.2.26: Suppose that we have a group of six people. each of whom own< a com...
Solutions for Chapter 2.2: MATRICES AND LINEAR TRANSFORMATIONS
Full solutions for Elementary Linear Algebra: A Matrix Approach | 2nd Edition
peA) = det(A - AI) has peA) = zero matrix.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Fast Fourier Transform (FFT).
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.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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.
A sequence of steps intended to approach the desired solution.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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 •
Outer product uv T
= column times row = rank one matrix.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.