 5.2.1: Label the following statements as true or false. (a) Any linear ope...
 5.2.2: For each of the following matrices A G Mnxn(R), test A for diagonal...
 5.2.3: For each of the following linear operators T on a vector space V, t...
 5.2.4: Prove the matrix version of the corollary to Theorem 5.5: If A G Mr...
 5.2.5: State and prove the matrix version of Theorem 5.6.
 5.2.6: (a) .Justify the test for diagonalizability and the method for diag...
 5.2.7: For find an expression for A", where n is an arbitrary positive int...
 5.2.8: Suppose that A G MnXn (F) has two distinct eigenvalues, Ai and A2, ...
 5.2.9: Let T be a lineal' operator on a finitedimensional vector space V,...
 5.2.10: Let T be a linear operator on a finitedimensional vector space V w...
 5.2.11: Let A be an n x n matrix that is similar to an upper triangular mat...
 5.2.12: Let T be an invertible linear operator on a finitedimensional vect...
 5.2.13: Let A G Mnxn (F). Recall from Exercise 14 of Section 5.1 that A and...
 5.2.14: Find the general solution to each system of differential equations....
 5.2.15: Let = x + y = 3x  y (b) x'i = 8x1 + 10x2 x'2 = 5xi  7x2 x1 = Xi x...
 5.2.16: Let C G Mmxn (/?), and let Y be an n x p matrix of differentiable f...
 5.2.17: Two linear operators T and U on a finitedimensional vector space V...
 5.2.18: Two linear operators T and U on a finitedimensional vector space V...
 5.2.19: Let T be a diagonalizable linear operator on a finitedimensional v...
 5.2.20: Let Wi, W2 ,..., Wfc be subspaces of a finitedimensional vector sp...
 5.2.21: Let V be a finitedimensional vector space with a basis 0, and let ...
 5.2.22: Let T be a linear operator on a finitedimensional vector space V, ...
 5.2.23: Let Wi, W2, Ki, K2 ,..., Kp, Mj, M2 ,..., M9 be subspaces of a vect...
Solutions for Chapter 5.2: Diagonalizability
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 5.2: Diagonalizability
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra , edition: 4. Chapter 5.2: Diagonalizability includes 23 full stepbystep solutions. Linear Algebra was written by and is associated to the ISBN: 9780130084514. Since 23 problems in chapter 5.2: Diagonalizability have been answered, more than 12117 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

CayleyHamilton Theorem.
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.

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Lucas numbers
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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.