 5.5.3.1: Prove that every third Fibonacci number in 0,1,1.2,3,... is even.
 5.5.4.1: Following the first example in this section, find the eigenvalues a...
 5.5.5.1: For the complex numbers 3+4i and 1i, (a) find their positions in th...
 5.5.6.1: If B is similar to A andC is similar to B, show thatC is similar to...
 5.5.1: Find the eigenvalues and eigenvectors, and the diagonalizing matrix...
 5.5.1.1: Find the eigenvalues and eigenvectors of the matrix A = 1 1 2 4 . V...
 5.5.2.1: Factor the following matrices into SS 1 : A = " 1 1 1 1# and A = " ...
 5.5.3.2: Bernadelli studied a beetle which lives three years only. and propa...
 5.5.4.2: For the previous matrix, write the general solution to du/dt = Au, ...
 5.5.5.2: What can you say about (a) the sum of a complex number and its conj...
 5.5.6.2: Describe in words all matrices that are similar to 1 0 0 1 , and fi...
 5.5.2: Find the determinants of A and A 1 if A = S " 1 2 0 2 # S 1 .
 5.5.1.2: With the same matrix A, solve the differential equation du/dt = Au,...
 5.5.2.2: Find the matrix A whose eigenvalues are 1 and 4, and whose eigenvec...
 5.5.3.3: For the Fibonacci matrix A = 1 1 1 0 , compute A 2 , A 3 , and A 4 ...
 5.5.4.3: Suppose the time direction is reversed to give the matrix A: du dt ...
 5.5.5.3: If x = 2+i and y = 1+3i, find x, xx, 1/x, and x/y. Check that the a...
 5.5.6.3: Explain why A is never similar to A+I
 5.5.3: If A has eigenvalues 0 and 1, corresponding to the eigenvectors " 1...
 5.5.1.3: If we shift to A 7I, what are the eigenvalues and eigenvectors and ...
 5.5.2.3: Find all the eigenvalues and eigenvectors of A = 1 1 1 1 1 1 1 1 1 ...
 5.5.3.4: Suppose each Gibonacci number Gk+2 is the average of the two previo...
 5.5.4.4: If P is a projection matrix, show from the infinite series that e P...
 5.5.5.4: Find a and b for the complex numbers a + ib at the angles = 30,60,9...
 5.5.6.4: Find a diagonal M, made up of 1s and 1s, to show that A = 2 1 1 2 1...
 5.5.4: In the previous problem, what will be the eigenvalues and eigenvect...
 5.5.1.4: Solve du/dt = Pu, when P is a projection: du dt = " 1 2 1 2 1 2 1 2...
 5.5.2.4: If a 3 by 3 upper triangular matrix has diagonal entries 1, 2, 7, h...
 5.5.3.5: Diagonalize the Fibonacci matrix by completing S 1 : " 1 1 1 0# = "...
 5.5.4.5: A diagonal matrix like = 1 0 0 2 satisfies the usual rule e (t+T) =...
 5.5.5.5: (a) If x = rei what are x 2 , x 1 , and x in polar coordinates? Whe...
 5.5.6.5: Show (if B is invertible) that BA is similar to AB.
 5.5.5: Does there exist a matrix A such that the entire family A + cI is i...
 5.5.1.5: Find the eigenvalues and eigenvectors of A = 3 4 2 0 1 2 0 0 0 and ...
 5.5.2.5: Which of these matrices cannot be diagonalized? A1 = " 2 2 2 2 # A2...
 5.5.3.6: The numbers k 1 and k 2 satisfy the Fibonacci rule Fk+2 = Fk+1 +Fk ...
 5.5.4.6: The higher order equation y 00 +y = 0 can be written as a firstord...
 5.5.5.6: Find the lengths and the inner product of x = " 24i 4i # and y = " ...
 5.5.6.6: (a) If CD = DC (and D is invertible), show that C is similar to C. ...
 5.5.6: Solve for both initial values and then find e At: du dt = " 3 1 1 3...
 5.5.1.6: Give an example to show that the eigenvalues can be changed when a ...
 5.5.2.6: (a) If A 2 = I, what are the possible eigenvalues of A? (b) If this...
 5.5.3.7: Lucas started with L0 = 2 and L1 = 1. The rule Lk+2 = Lk+1 +Lk is t...
 5.5.4.7: Convert y 00 = 0 to a firstorder system du/dt = Au: d dt " y y 0 #...
 5.5.5.7: Write out the matrix A H and compute C = A HA if A = " 1 i 0 i 0 1#...
 5.5.6.7: Consider any A and a Givens rotation M in the 12 plane: A = a b c d...
 5.5.7: Would you prefer to have interest compounded quarterly at 40% per y...
 5.5.1.7: Suppose that is an eigenvalue of A, and x is its eigenvector: Ax = ...
 5.5.2.7: If A = 4 3 1 2 , find A 100 by diagonalizing A.
 5.5.3.8: Suppose there is an epidemic in which every month half of those who...
 5.5.4.8: Suppose the rabbit population r and the wolf population w are gover...
 5.5.5.8: (a) With the preceding A, use elimination to solve Ax = 0. (b) Show...
 5.5.6.8: What matrix M changes the basis V1 = (1,1), V2 = (1,4) to the basis...
 5.5.8: True or false (with counterexample if false): (a) If B is formed fr...
 5.5.1.8: Show that the determinant equals the product of the eigenvalues by ...
 5.5.2.8: Suppose A = uvT is a column times a row (a rank1 matrix). (a) By m...
 5.5.3.9: Suppose there is an epidemic in which every month half of those who...
 5.5.4.9: Decide the stability of u 0 = Au for the following matrices: (a) A ...
 5.5.5.9: (a) How is the determinant of A H related to the determinant of A? ...
 5.5.6.9: For the same two bases, express the vector (3,9) as a combination c...
 5.5.9: What happens to the Fibonacci sequence if we go backward in time, a...
 5.5.1.9: Show that the trace equals the sum of the eigenvalues, in two steps...
 5.5.2.9: Show by direct calculation that AB and BA have the same trace when ...
 5.5.3.10: Find the limiting values of yk and k (k ) if yk+1 = .8yk +.3zk y0 =...
 5.5.4.10: Decide on the stability or instability of dv/dt = w, dw/dt = v. Is ...
 5.5.5.10: (a) How many degrees of freedom are there in a real symmetric matri...
 5.5.6.10: Confirm the last exercise: If V1 = m11v1 + m21v2 and V2 = m12v1 + m...
 5.5.1.10: (a) Construct 2 by 2 matrices such that the eigenvalues of AB are n...
 5.5.2.10: Suppose A has eigenvalues 1, 2, 4. What is the trace of A 2 ? What ...
 5.5.3.11: (a) From the fact that column 1 + column 2 = 2(column 3), so the co...
 5.5.4.11: From their trace and determinant, at what time t do the following m...
 5.5.5.11: Write P, Q and R in the form 1x1x H 1 +2x2x H 2 of the spectral the...
 5.5.6.11: If the transformation T is a reflection across the 45 line in the p...
 5.5.11: If P is the matrix that projects R n onto a subspace S, explain why...
 5.5.1.11: The eigenvalues of A equal the eigenvalues of A T . This is because...
 5.5.2.11: If the eigenvalues of A are 1, 1, 2, which of the following are cer...
 5.5.3.12: Suppose there are three major centers for MoveItYourself trucks. ...
 5.5.4.12: Find the eigenvalues and eigenvectors for du dt = Au = 0 3 0 3 0 4 ...
 5.5.5.12: Give a reason if true or a counterexample if false: (a) If A is Her...
 5.5.6.12: The identity transformation takes every vector to itself: T x = x. ...
 5.5.12: Show that every matrix of order > 1 is the sum of two singular matr...
 5.5.1.12: Find the eigenvalues and eigenvectors of A = " 3 4 4 3 # and A = " ...
 5.5.2.12: Suppose the only eigenvectors of A are multiples of x = (1,0,0). Tr...
 5.5.3.13: (a) In what range of a and b is the following equation a Markov pro...
 5.5.4.13: For the skewsymmetric equation du dt = Au = 0 c b c 0 a b a 0 u1 u...
 5.5.5.13: Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2. (a...
 5.5.6.13: The derivative of a+bx+cx2 is b+2cx+0x 2 . (a) Write the 3 by 3 mat...
 5.5.13: (a) Show that the matrix differential equation dX/dt = AX + XB has ...
 5.5.1.13: If B has eigenvalues 1, 2, 3, C has eigenvalues 4, 5, 6, and D has ...
 5.5.2.13: Diagonalize the matrix A = 5 4 4 5 and find one of its square roots...
 5.5.3.14: Multinational companies in the Americas, Asia, and Europe have asse...
 5.5.4.14: What are the eigenvalues and frequencies , and the general solution...
 5.5.5.14: In the list below, which classes of matrices contain A and which co...
 5.5.6.14: Show that every number is an eigenvalue for T f(x) = d f /dx, but t...
 5.5.14: If the eigenvalues of A are 1 and 3 with eigenvectors (5,2) and (2,...
 5.5.1.14: Find the rank and all four eigenvalues for both the matrix of ones ...
 5.5.2.14: Suppose the eigenvector matrix S has S T = S 1 . Show that A = SS 1...
 5.5.3.15: If A is a Markov matrix, show that the sum of the components of Ax ...
 5.5.4.15: Solve the secondorder equation d 2u dt2 = " 5 1 1 5 # u with u(0) ...
 5.5.5.15: What is the dimension of the space S of all n by n real symmetric m...
 5.5.6.15: On the space of 2 by 2 matrices, let T be the transformation that t...
 5.5.15: Find the eigenvalues and eigenvectors of A = 0 i 0 i 1 i 0 i 0 . Wh...
 5.5.1.15: What are the rank and eigenvalues when A and C in the previous exer...
 5.5.2.15: Factor these two matrices into A = SS 1 : A = " 1 2 0 3# and A = " ...
 5.5.3.16: The solution to du/dt = Au = 0 1 1 0 u (eigenvalues i and i) goes a...
 5.5.4.16: In most applications the secondorder equation looks like Mu00+Ku =...
 5.5.5.16: Write one significant fact about the eigenvalues of each of the fol...
 5.5.6.16: (a) Find an orthogonal Q so that Q 1AQ = if A = 1 1 1 1 1 1 1 1 1 a...
 5.5.16: By trying to solve " a b c d#"a b c d# = " 0 1 0 0# = A show that A...
 5.5.1.16: If A is the 4 by 4 matrix of ones, find the eigenvalues and the det...
 5.5.2.16: If A = SS 1 then A 3 = ( )( )( ) and A 1 = ( )( )( ).
 5.5.3.17: What values of produce instability in vn+1 = (vn +wn), wn+1 = (vn +wn)
 5.5.4.17: With a friction matrix F in the equation u 00 +Fu0 Au = 0, substitu...
 5.5.5.17: Show that if U and V are unitary, so is UV. Use the criterion U HU ...
 5.5.6.17: Prove that every unitary matrix A is diagonalizable, in two steps: ...
 5.5.17: (a) Find the eigenvalues and eigenvectors of A = h 0 4 1 4 0 i . (b...
 5.5.1.17: Choose the third row of the companion matrix A = 0 1 0 0 0 1 so tha...
 5.5.2.17: If A has 1 = 2 with eigenvector x1 = 1 0 and 2 = 5 with x2 = 1 1 , ...
 5.5.3.18: Find the largest a, b, c for which these matrices are stable or neu...
 5.5.4.18: For equation (16) in the text, with = 1 and 3, find the motion if t...
 5.5.5.18: Show that a unitary matrix has detU = 1, but possibly detU is dif...
 5.5.6.18: Find a normal matrix (NNH = N HN) that is not Hermitian, skewHermi...
 5.5.18: True or false, with reason if true and counterexample if false: (a)...
 5.5.1.18: Suppose A has eigenvalues 0, 3, 5 with independent eigenvectors u, ...
 5.5.2.18: Suppose A = SS 1 . What is the eigenvalue matrix for A+2I? What is ...
 5.5.2.19: True or false: If the n columns of S (eigenvectors of A) are indepe...
 5.5.3.19: Multiplying term by term, check that (IA)(I + A + A2 + ) = I. This ...
 5.5.4.19: Every 2 by 2 matrix with trace zero can be written as A = " a b+c b...
 5.5.5.19: Find a third column so that U is unitary. How much freedom in colum...
 5.5.6.19: Suppose T is a 3 by 3 upper triangular matrix, with entriesti j. Co...
 5.5.19: If K is a skewsymmetric matrix, show that Q = (I K)(I +K) 1 is an ...
 5.5.1.19: The powers A k of this matrix A approaches a limit as k : A = " .8 ...
 5.5.2.20: If the eigenvectors of A are the columns of I, then A is a matrix. ...
 5.5.3.20: For A = 0 .2 0 .5 , find the powers A k (including A 0 ) and show e...
 5.5.4.20: By backsubstitution or by computing eigenvectors, solve du dt = 1 ...
 5.5.5.20: Diagonalize the 2 by 2 skewHermitian matrix K = i i i i , whose en...
 5.5.6.20: If N is normal, show that kNxk = kN Hxk for every vector x. Deduce ...
 5.5.1.20: Find the eigenvalues and the eigenvectors of these two matrices: A ...
 5.5.2.21: Describe all matrices S that diagonalize this matrix A: A = " 4 0 1...
 5.5.3.21: Explain by mathematics or economics why increasing the consumption ...
 5.5.4.21: Find s and xs so that u = e t x solves du dt = " 4 3 0 1# u. What c...
 5.5.5.21: Describe all 3 by 3 matrices that are simultaneously Hermitian, uni...
 5.5.6.21: Prove that a matrix with orthonormal eigenvectors must be normal, a...
 5.5.21: If M is the diagonal matrix with entries d, d 2 , d 3 , what is M1A...
 5.5.1.21: Compute the eigenvalues and eigenvectors of A and A 1 : A = " 0 2 2...
 5.5.2.22: Write the most general matrix that has eigenvectors 1 1 and 1 1 .
 5.5.3.22: What are the limits as k (the steady states) of the following? h .4...
 5.5.4.22: Solve for u(t) = (y(t),z(t)) by backsubstitution: First solve dz d...
 5.5.5.22: Every matrix Z can be split into a Hermitian and a skewHermitian p...
 5.5.6.22: Find a unitary U and triangular T so that U 1AU = T, for A = " 5 3 ...
 5.5.22: If A 2 = I, what are the eigenvalues of A? If A is a real n by n ma...
 5.5.1.22: Compute the eigenvalues and eigenvectors of A and A 2 : A = " 1 3 2...
 5.5.2.23: Find the eigenvalues of A and B and A+B: A = " 1 0 1 1# , B = " 1 1...
 5.5.3.23: Diagonalize A and compute S kS 1 to prove this formula for A k : A ...
 5.5.4.23: Find A to change y 00 = 5y 0 +4y into a vector equation for u(t) = ...
 5.5.5.23: Show that the columns of the 4 by 4 Fourier matrix F in Example 5 a...
 5.5.6.23: If A has eigenvalues 0, 1, 2, what are the eigenvalues of A(AI)(A2I)?
 5.5.23: If Ax = 1x and A T y = 2y (all real), show that x T y = 0.
 5.5.1.23: (a) If you know x is an eigenvector, the way to find is to . (b) If...
 5.5.2.24: Find the eigenvalues of A, B, AB, and BA: A = " 1 0 1 1# , B = " 1 ...
 5.5.3.24: Diagonalize B and compute S kS 1 to prove this formula for B k : B ...
 5.5.4.24: A door is opened between rooms that hold v(0) = 30 people and w(0) ...
 5.5.5.24: For the permutation of Example 6, write out the circulant matrix C ...
 5.5.6.24: (a) Show by direct multiplication that every triangular matrix T, s...
 5.5.24: A variation on the Fourier matrix is the sine matrix: S = 1 2 sin s...
 5.5.1.24: What do you do to Ax = x, in order to prove (a), (b), and (c)? (a) ...
 5.5.2.25: True or false: If the eigenvalues of A are 2, 2, 5, then the matrix...
 5.5.3.25: The eigenvalues of A are 1 and 9, the eigenvalues of B are 1 and 9:...
 5.5.4.25: Reverse the diffusion of people in to du/dt = Au: dv dt = vw and dw...
 5.5.5.25: For a circulant C = FF 1 , why is it faster to multiply by F 1 , th...
 5.5.6.25: The characteristic polynomial of A = a b c d is 2 (a+d) + (ad bc). ...
 5.5.25: (a) Find a nonzero matrix N such that N 3 = 0. (b) If Nx = x, show ...
 5.5.1.25: From the unit vector u = 1 6 , 1 6 , 3 6 , 5 6 , construct the rank...
 5.5.2.26: If the eigenvalues of A are 1 and 0, write everything you know abou...
 5.5.3.26: If A and B have the same s with the same full set of independent ei...
 5.5.4.26: The solution to y 00 = 0 is a straight line y = C +Dt. Convert to a...
 5.5.5.26: Find the lengths of u = (1+i,1i,1+2i) and v = (i,i,i). Also find u ...
 5.5.6.26: If ai j = 1 above the main diagonal and ai j = 0 elsewhere, find th...
 5.5.26: (a) Find the matrix P = aaT/a Ta that projects any vector onto the ...
 5.5.1.26: Solve det(QI) = 0 by the quadratic formula, to reach = cos isin: Q ...
 5.5.2.27: Complete these matrices so that detA = 25. Then trace = 10, and = 5...
 5.5.3.27: Suppose A and B have the same full set of eigenvectors, so that A =...
 5.5.4.27: Substitute y = e t into y 00 = 6y 0 9y to show that = 3 is a repeat...
 5.5.5.27: Prove that A HA is always a Hermitian matrix, Compute A HA and AAH:...
 5.5.6.27: Show, by trying for an M and failing, that no two of the three Jord...
 5.5.27: Suppose the first row of A is 7, 6 and its eigenvalues are i, i. Fi...
 5.5.1.27: Every permutation matrix leaves x = (1,1,...,1) unchanged. Then = 1...
 5.5.2.28: The matrix A = 3 1 0 3 is not diagonalizable because the rank of A ...
 5.5.3.28: (a) When do the eigenvectors for = 0 span the nullspace N(A)? (b) W...
 5.5.4.28: Figure out how to write my00 +by0 +ky = 0 as a vector equation Mu0 ...
 5.5.5.28: If Az = 0, then A HAz = 0. If A HAz = 0, multiply by z H to prove t...
 5.5.6.28: Solve u 0 = Ju by backsubstitution, solving first for u2(t): du dt...
 5.5.28: (a) For which numbers c and d does A have real eigenvalues and orth...
 5.5.1.28: If A has 1 = 4 and 2 = 5, then det(AI) = ( 4)( 5) = 2 9 +20. Find t...
 5.5.2.29: A k = S kS 1 approaches the zero matrix as k if and only if every h...
 5.5.3.29: The powers A k approach zero if all i  < 1, and they blow up if a...
 5.5.4.29: Figure out how to write my00 +by0 +ky = 0 as a vector equation Mu0 ...
 5.5.5.29: When you multiply a Hermitian matrix by a real number c, is cA stil...
 5.5.6.29: Compute A 10 and e A if A = MJM1 : A = " 14 9 16 10# = " 3 2 4 3 #"...
 5.5.29: If the vectors x1 and x2 are in the columns of S, what are the eige...
 5.5.1.29: A 3 by 3 matrix B is known to have eigenvalues 0, 1, 2, This inform...
 5.5.2.30: (Recommended) Find and S to diagonalize A in 29. What is the limit ...
 5.5.4.30: A particular solution to du/dt = Aub is up = A 1b, if A is invertib...
 5.5.5.30: Which classes of matrices does P belong to: orthogonal, invertible,...
 5.5.6.30: Show that A and B are similar by finding M so that B = M1AM: (a) A ...
 5.5.1.30: Choose the second row of A = [ 0 1 ] so that A has eigenvalues 4 an...
 5.5.2.31: Find and S to diagonalize B in 29. What is B 10u0 for these u0? u0 ...
 5.5.4.31: If c is not an eigenvalue of A, substitute u = e ctv and find v to ...
 5.5.5.31: Compute P 2 , P 3 , and P 100 in 30. What are the eigenvalues of P?
 5.5.6.31: Which of these matrices A1 to A6 are similar? Check their eigenvalu...
 5.5.1.31: Choose a, b, c, so that det(AI) = 9 3 . Then the eigenvalues are 3,...
 5.5.2.32: Diagonalize A and compute S kS 1 to prove this formula for A k : A ...
 5.5.4.32: Find a matrix A to illustrate each of the unstable regions in Figur...
 5.5.5.32: Find the unit eigenvectors of P in 30, and put them into the column...
 5.5.6.32: There are sixteen 2 by 2 matrices whose entries are 0s and 1s. Simi...
 5.5.1.32: Construct any 3 by 3 Markov matrix M: positive entries down each co...
 5.5.2.33: Diagonalize B and compute S kS 1 to prove this formula for B k : B ...
 5.5.4.33: Write five terms of the infinite series for e At. Take the t deriva...
 5.5.5.33: Write down the 3 by 3 circulant matrix C = 2I + 5P + 4P 2 . It has ...
 5.5.6.33: (a) If x is in the nullspace of A, show that M1 x is in the nullspa...
 5.5.1.33: Find three 2 by 2 matrices that have 1 = 2 = 0. The trace is zero a...
 5.5.2.34: Suppose that A = SS 1 . Take determinants to prove that detA = 12 n...
 5.5.4.34: The matrix B = 0 1 0 0 has B 2 = 0. Find e Bt from a (short) infini...
 5.5.5.34: If U is unitary and Q is a real orthogonal matrix, show that U 1 is...
 5.5.6.34: If A and B have the exactly the same eigenvalues and eigenvectors, ...
 5.5.1.34: This matrix is singular with rank 1. Find three s and three eigenve...
 5.5.2.35: The trace of S times S 1 equals the trace of S 1 times S. So the tr...
 5.5.4.35: Starting from u(0), the solution at time T is e AT u(0). Go an addi...
 5.5.5.35: Diagonalize A (real s) and K (imaginary s) to reach UU H: A = " 0 1...
 5.5.6.35: By direct multiplication, find J 2 and J 3 when J = " c 1 0 c # . G...
 5.5.1.35: Suppose A and B have the same eigenvalues 1,...,n with the same ind...
 5.5.2.36: If A = SS 1 , diagonalize the block matrix B = A 0 0 2A . Find its ...
 5.5.4.36: Write A = 1 1 0 0 in the form SS 1 . Find e At from SetS 1 .
 5.5.5.36: Diagonalize this orthogonal matrix to reach Q = UU H. Now all s are...
 5.5.6.36: If J is the 5 by 5 Jordan block with = 0, find J 2 and count its ei...
 5.5.1.36: (Review) Find the eigenvalues of A, B, and C: A = 1 2 3 0 4 5 0 0 6...
 5.5.2.37: Consider all 4 by 4 matrices A that are diagonalized by the same fi...
 5.5.4.37: f A 2 = A, show that the infinite series produces e At = I + (e t 1...
 5.5.5.37: Diagonalize this unitary matrix V to reach V = UU H. Again all  =...
 5.5.6.37: The text solved du/dt = Ju for a 3 by 3 Jordan block J. Add a fourt...
 5.5.1.37: When a+b = c+d, show that (1,1) is an eigenvector and find both eig...
 5.5.2.38: Suppose A 2 = A. On the left side A multiplies each column of A. Wh...
 5.5.4.38: Generally e A e B is different from e B e A . They are both differe...
 5.5.5.38: If v1,...,vn is an orthonormal basis for C n , the matrix with thos...
 5.5.6.38: These Jordan matrices have eigenvalues 0, 0, 0, 0. They have two ei...
 5.5.1.38: When P exchanges rows 1 and 2 and columns 1 and 2, the eigenvalues ...
 5.5.2.39: Suppose Ax = x. If = 0, then x is in the nullspace. If 6= 0, then x...
 5.5.4.39: Write A = 1 1 0 3 as SS 1 . Multiply SetS 1 to find the matrix expo...
 5.5.5.39: The functions e ix and e ix are orthogonal on the interval 0 x 2 be...
 5.5.6.39: Prove in three steps that A T is always similar to A (we know that ...
 5.5.1.39: Challenge problem: Is there a real 2 by 2 matrix (other than I) wit...
 5.5.2.40: Substitute A = SS 1 into the product (A 1I)(A 2I)(A nI) and explain...
 5.5.4.40: Put A = 1 3 0 0 into the infinite series to find e At. First comput...
 5.5.5.40: The vectors v = (1,i,1), w = (i,1,0) and z = are an orthogonal basi...
 5.5.6.40: Which pairs are similar? Choose a, b, c, d to prove that the other ...
 5.5.1.40: There are six 3 by 3 permutation matrices P. What numbers can be th...
 5.5.2.41: Test the CayleyHamilton Theorem on Fibonaccis matrix A = 1 1 1 0 ....
 5.5.4.41: Give two reasons why the matrix exponential e At is never singular:...
 5.5.5.41: If A = R+iS is a Hermitian matrix, are the real matrices R and S sy...
 5.5.6.41: True or false, with a good reason: (a) An invertible matrix cant be...
 5.5.2.42: If A = a b c d , then det(AI) is ( a)( d). Check the CayleyHamilto...
 5.5.4.42: Find a solution x(t), y(t) of the first system that gets large as t...
 5.5.5.42: The (complex) dimension of C n is . Find a nonreal basis for C n .
 5.5.6.42: Prove that AB has the same eigenvalues as BA.
 5.5.2.43: If A = 1 0 0 2 and AB = BA, show that B = a b c d is also diagonal....
 5.5.4.43: From this general solution to du/dt = Au, find the matrix A: u(t) =...
 5.5.5.43: Describe all 1 by 1 matrices that are Hermitian and also unitary. D...
 5.5.6.43: If A is 6 by 4 and B is 4 by 6, AB and BA have different sizes. Nev...
 5.5.2.44: If A is 5 by 5. then ABBA = zero matrix gives 25 equations for the ...
 5.5.5.44: How are the eigenvalues of A H (square matrix) related to the eigen...
 5.5.6.44: Why is each of these statements true? (a) If A is similar to B, the...
 5.5.2.45: Find the eigenvalues and eigenvectors for both of these Markov matr...
 5.5.5.45: If u Hu = 1, show that I 2uuH is Hermitian and also unitary. The ra...
 5.5.5.46: If A+iB is a unitary matrix (A and B are real), show that Q = A B B...
 5.5.5.47: If A+iB is a Hermitian matrix (A and B are real), show that A B B A...
 5.5.5.48: Prove that the inverse of a Hermitian matrix is again a Hermitian m...
 5.5.5.49: Diagonalize this matrix by constructing its eigenvalue matrix and i...
 5.5.5.50: A matrix with orthonormal eigenvectors has the form A = UU 1 = UU H...
Solutions for Chapter 5: Eigenvalues and Eigenvectors
Full solutions for Linear Algebra and Its Applications,  4th Edition
ISBN: 9780030105678
Solutions for Chapter 5: Eigenvalues and Eigenvectors
Get Full SolutionsLinear Algebra and Its Applications, was written by and is associated to the ISBN: 9780030105678. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5: Eigenvalues and Eigenvectors includes 278 full stepbystep solutions. Since 278 problems in chapter 5: Eigenvalues and Eigenvectors have been answered, more than 11178 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications,, edition: 4.

Acceleration due to gravity
g ? 32 ft/sec2 ? 9.8 m/sec

Amplitude
See Sinusoid.

Cardioid
A limaçon whose polar equation is r = a ± a sin ?, or r = a ± a cos ?, where a > 0.

Cone
See Right circular cone.

Equilibrium price
See Equilibrium point.

Exponential regression
A procedure for fitting an exponential function to a set of data.

Firstdegree equation in x , y, and z
An equation that can be written in the form.

Grapher or graphing utility
Graphing calculator or a computer with graphing software.

Intercept
Point where a curve crosses the x, y, or zaxis in a graph.

Local extremum
A local maximum or a local minimum

Ordered set
A set is ordered if it is possible to compare any two elements and say that one element is “less than” or “greater than” the other.

Positive numbers
Real numbers shown to the right of the origin on a number line.

Present value of an annuity T
he net amount of your money put into an annuity.

Radicand
See Radical.

Random behavior
Behavior that is determined only by the laws of probability.

Range (in statistics)
The difference between the greatest and least values in a data set.

symmetric about the xaxis
A graph in which (x, y) is on the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ?, ?) is on the graph whenever (r, ?) is

Tangent
The function y = tan x

Unit vector
Vector of length 1.

Vector equation for a line in space
The line through P0(x 0, y0, z0) in the direction of the nonzero vector V = <a, b, c> has vector equation r = r0 + tv , where r = <x,y,z>.