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

Expert Solution

Given that U is a square matrix with n rows and n columns.

It is a uppee triangular matrix. It means all the elements below the diagonal of the matrix are zeros. Rank of matrix U is n.

For the system Ux=b, as U has n rows, we have n no.of values of x from x1,x2,x3,......,xn.

Given that on computation resource, it take one micro second for one arithmetic operation. Here, in a computation resource, to solve a problem, it may require n number of steps to solve the system.

In this case, time taken is number of steps multiplied by 1micro second.

As Ux=b has n values of x, we will get n number of equations fron Ux=b to find n values of x.

So, time to solve the system = number of equations * time for each srep.

I.e total time = (n)*(1micro sec) = n micro seconds

For a lower triangular matrix, all the elements above the diagonal are zeros. To solve the system Ux=b, we get n equations to solve to find n values of x.

So for lower triangular matrix with the system Ux=b, total time to solve is same as for the upoer triangular matrix.

Hence, time for solving Ux=b, where U is either upper triangular or lower triangular matrix is n microseconds.

Leave a comment for any queries. Upvote if u like it.

Thank you.

Colby Messinger answered 1 year ago

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.

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?

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.

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

Let “ ·n” be multiplication modulo n, and consider the set Un = { [a] ∈...

Let “ ·n” be multiplication modulo n, and consider the set Un = { [a] ∈ Zn | there is a [b] ∈ Zn with [a] ·n [b] = [1]} (a) Show that (Un, ·n ) is a group. (b) Write down the Cayley table for U5. Hint: |U5| = 4. (c) Write down the Cayley table for U12. Hint: |U12| = 4.

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6 0 −4 , A5 A5 =

Question

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

Solutions

Expert Solution

Related Solutions

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

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

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

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

Let “ ·n” be multiplication modulo n, and consider the set Un = { [a] ∈...

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let (Un, U, n>1) be asequence of random variables such that Un and U are independent,...

Let A be an m × n matrix and B be an m × p matrix....

Let v1 be an eigenvector of an n×n matrix A corresponding to λ1, and let v2,...