 9.4.1: For Exercises 18, find the change of basis matrix from G to HG = {2...
 9.4.2: For Exercises 18, find the change of basis matrix from G to HG = {x...
 9.4.3: For Exercises 18, find the change of basis matrix from G to HG =3 2...
 9.4.4: For Exercises 18, find the change of basis matrix from G to HG = {x...
 9.4.5: For Exercises 18, find the change of basis matrix from G to HG = {x...
 9.4.6: For Exercises 18, find the change of basis matrix from G to HG =6 1...
 9.4.7: For Exercises 18, find the change of basis matrix from G to HG = {7...
 9.4.8: For Exercises 18, find the change of basis matrix from G to HG = {x...
 9.4.9: For Exercises 912, B is the matrix of T : V V with respect toa basi...
 9.4.10: For Exercises 912, B is the matrix of T : V V with respect toa basi...
 9.4.11: For Exercises 912, B is the matrix of T : V V with respect toa basi...
 9.4.12: For Exercises 912, B is the matrix of T : V V with respect toa basi...
 9.4.13: For Exercises 1316, B is the matrix of T : V V with respectto the b...
 9.4.14: For Exercises 1316, B is the matrix of T : V V with respectto the b...
 9.4.15: For Exercises 1316, B is the matrix of T : V V with respectto the b...
 9.4.16: For Exercises 1316, B is the matrix of T : V V with respectto the b...
 9.4.17: For Exercises 1720, determine if A and B are similar matrices.A =1 ...
 9.4.18: For Exercises 1720, determine if A and B are similar matrices.A = 2...
 9.4.19: For Exercises 1720, determine if A and B are similar matrices.A =1 ...
 9.4.20: For Exercises 1720, determine if A and B are similar matrices.A =12...
 9.4.21: FIND AN EXAMPLE For Exercises 2124, find an example thatmeets the g...
 9.4.22: FIND AN EXAMPLE For Exercises 2124, find an example thatmeets the g...
 9.4.23: FIND AN EXAMPLE For Exercises 2124, find an example thatmeets the g...
 9.4.24: FIND AN EXAMPLE For Exercises 2124, find an example thatmeets the g...
 9.4.25: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.26: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.27: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.28: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.29: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.30: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.31: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.32: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.33: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.34: TRUE OR FALSE For Exercises 2534, determine if the statementis true...
 9.4.35: Suppose that A and B are similar matrices, related by A =S11 B S1, ...
 9.4.36: Prove that similarity of matrices is transitive: if A is similar to...
 9.4.37: Suppose that A and B are both diagonalizable matrices thathave the ...
 9.4.38: Prove that if A and B are similar matrices, then so are Akand Bk .
 9.4.39: Prove that if A and B are similar matrices, then so are ATand BT .
 9.4.40: Suppose that A and B are similar matrices and that A isinvertible. ...
 9.4.41: C For Exercises 4144, determine if the given matrices are similar.A...
 9.4.42: C For Exercises 4144, determine if the given matrices are similar.A...
 9.4.43: C For Exercises 4144, determine if the given matrices are similar.A...
 9.4.44: C For Exercises 4144, determine if the given matrices are similar.A...
Solutions for Chapter 9.4: Similarity
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 9.4: Similarity
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.)

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

Column picture of Ax = b.
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).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Norm
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.

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.