et A be a 157 x 157 upper-triangular matrix. Suppose that every diagonal entry of A...

et A be a 157 x 157 upper-triangular matrix. Suppose that every diagonal entry of A is 1 and that there is at least one nonzero off-diagonal entry in A. Is A diagonalizable? Explain how you can answer this question mentally, with no non-trivial calculations.

Expert Solution

Colby Messinger answered 2 years ago

1. Let ??(F), ??(F) and ??(F) denote the spaces of strictly upper triangular, diagonal, and strictly...

1. Let ??(F), ??(F) and ??(F) denote the spaces of strictly upper triangular, diagonal, and strictly lower triangular ?×? matrices over F, respectively. Show that ??(F) = ??(F)⊕??(F)⊕??(F). Compute the dimensions of these spaces and show they sum 2 up to ?. 2. Is the set {?3 +2?2,−?2 +3?+1,?3 −?2 +2?−1} in P3(R) linearly independent? Is it a basis?

1) Let A be nxn matrix and Ax=b, if we need change A to Upper triangular...

1) Let A be nxn matrix and Ax=b, if we need change A to Upper triangular matrix using Gaussian Elimination, how many additions/subtraction operations are involved? how many multiplication/division operations are involved? 2) Once we got the upper triangular matrix, now we need to apply back-substitution, how many additions/subtraction operations are involved? how many multiplication/division operations are involved?

MATLAB Function: (Backward substitution) Write a function that takes an upper triangular matrix U, and a...

MATLAB Function: (Backward substitution) Write a function that takes an upper triangular matrix U, and a vector b as input and returns the solution of the linear system U x = b. Use this function in the case when U is made of the numbers [(3,2,1),(0,5,4),(0,0,6)], and b = [1,1,−6]^T.

for matrices, what is the difference between row reduced echelon form and an upper triangular matrix?

Let Un×n be an upper triangular matrix of rank n. If any arithmetic operation takes 1µ...

Let Un×n be an upper triangular matrix of rank n. If any arithmetic operation takes 1µ second on a computing resource, compute the time taken to solve the system Ux = b, assuming it has a unique solution. What would be the time taken if Un×n is lower triangular

Let A be a real n × n matrix, and suppose that every leading principal submatrix...

Let A be a real n × n matrix, and suppose that every leading principal submatrix ofA of order k < n is nonsingular. Show that A has an LU-factorisation.

Suppose that A is a 6 x 6 matrix that can be written as a product...

Suppose that A is a 6 x 6 matrix that can be written as a product of matrices A = BC where B is 6 x 4 and C is 4 x 6. Prove that A is not invertible. An economist builds a Leontief input-output model for the interaction between the mining and energy sectors of a local economy using the following assumptions: In order to produce 1 million dollars of output, the mining sector requires 0.1 million dollars of...

Suppose A is an mxn matrix of real numbers and x in an nx1 column vector....

Suppose A is an mxn matrix of real numbers and x in an nx1 column vector. a.) suppose Ax=0. Show that ATAx=0. b.)Suppose ATAx=0. show Ax=0. c.) by part a and b, we can conclude that Nul(A) = Nul(ATA), and thus dim(Nul A) = dim(Nul(ATA)), and thus nullity(A) = nullity(ATA). prove the columns of A are linearly independent iff ATA is invertible.

suppose that for every pair of disjoint closed sets A and B of X there exist...

suppose that for every pair of disjoint closed sets A and B of X there exist a countinous function f:X-> {0,1} such that f(A)={0} and f(B)={1}. show one point set is closed in X.

Problem 4: Suppose M is a random matrix, and x is a deterministic (fixed) column vector....

Problem 4: Suppose M is a random matrix, and x is a deterministic (fixed) column vector. Show that E[x' M x] = x' E[M] x, where x' denotes the transpose of x.

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