- 1.2.1: LeI A = [~ and c~ (a) Whatisa I2.an.(I2l'! (b) Whatishll .b31 '! (e...
- 1.2.2: Determine the incidence matrix associated with each of the followin...
- 1.2.3: For each of the following incidence matrices. construct a graph. La...
- 1.2.4: If [" +b c - d <+d] [4 (/ - b = [0
- 1.2.5: If [a + 2b 2c + d 2a - bH4 c - 2d 4 find a, b, c . and d.
- 1.2.6: III xercise~' 6 rhrvugh 9, lei A = [~ 2 :1 B~ [i c ~ [~ - I lJ D = ...
- 1.2.7: III xercise~' 6 rhrvugh 9, lei A = [~ 2 :1 B~ [i c ~ [~ - I lJ D = ...
- 1.2.8: III xercise~' 6 rhrvugh 9, lei A = [~ 2 :1 B~ [i c ~ [~ - I lJ D = ...
- 1.2.9: III xercise~' 6 rhrvugh 9, lei A = [~ 2 :1 B~ [i c ~ [~ - I lJ D = ...
- 1.2.10: Is the matnx . [' 0 0]. 2 a lmear combmatlOn .. of the matn. - OJ 1...
- 1.2.11: Is the matnx . [' 0 - 3 ']. a hnear comblllatlOn .. of the ma. [ , ...
- 1.2.12: Let 2 2 2 If). is a real number. compute AlJ - A. o ,
- 1.2.13: If A is an /I x /I matrix. what are the entries on the main diagona...
- 1.2.14: Explain why every incidence matrix A associated with a graph is the...
- 1.2.15: Let the /I x /I matrix A be equal to A T. Briefly describe lhe pall...
- 1.2.16: If x is an /I-vector. show that x + 0 = x.
- 1.2.17: Show lhat the summation notation satisfies the following properties...
- 1.2.18: ShOWlh"t(~U' )~~(tu )
- 1.2.19: Identify the following expressions as true or false. If true. prove...
- 1.2.20: A large steel manufacturer. who has 2000 employees. lists each empl...
- 1.2.21: A brokerage finn records the high and low values of the price of IB...
- 1.2.22: For the software you are using. determine the commands to enter a m...
- 1.2.23: Determine whether the software you are llsing includes a computer a...
- 1.2.24: For the software you are using, detennine whether there IS a comman...
Solutions for Chapter 1.2: Matrices
Full solutions for Elementary Linear Algebra with Applications | 9th Edition
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Invert A by row operations on [A I] to reach [I A-I].
A symmetric matrix with eigenvalues of both signs (+ and - ).
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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.
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.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).