Homework Answers
Here is the matrix mentioned in the problem: 7) Change (23 pts) Diagonalization of a matrix....
Here is the matrix mentioned in the problem:
7) Change (23 pts) Diagonalization of a matrix. For the matrix from the last problem. a) (6 pts) Find the eigenvalues of A b) (6 pts) Find the eigenvectors of A, Choose x1=1 for the first element of both eigenvectors. c) (4 pts) Is this system stable? What type of damping does it have? d) (2 pts) Find the diagonal matrix D corresponding to A. e) (1 pts) Find D3, use actual...
3. Suppose A is a real square n x n matrix with SVD given by A USVT Using MATLAB's eig and svd, i...
3. Suppose A is a real square n x n matrix with SVD given by A USVT Using MATLAB's eig and svd, investigate how the eigenvalues and eigen- vectors of the real symmetric matrix AT 0 depend on 2, U and V. Try a random matrix with n 2 to get started. Once you see the relationship, state it carefully, without proof. 4. (This is a continuation of the previous question.) Prove the property that you observed in the previous...
Theory: A vector with nonnegative entries is called a probability vector if the sum of its entries is 1. A square matrix...
Theory: A vector with nonnegative entries is called a probability vector if the sum of its entries is 1. A square matrix is called right stochastic matrix if its rows are probability vectors; a square matrix is called a left stochastic matrix if its columns are probability vectors; and a square matrix is called a doubly stochastic matrix if both the rows and the columns are probability vectors. **Write a MATLAB function function [S1,S2,P]=stochastic(A) which accepts a square matrix A...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Question 3 (a) Use the defining equation for eigenvalues and eigenvectors to prove that if matrix...
Question 3 (a) Use the defining equation for eigenvalues and eigenvectors to prove that if matrix A has the unique eigenvalue a, then the matrix A-al has 0 as an eigenvalue (b) Show that if matrix B has the eigenvalue then the matri B has 2 as an eigenvalue. (c) Use the defining equation to show that if the matrix C is invertible, then C cannot have zero as an eigenvalue. (Hint: No eigenvector Xcan be the zero vector. So...
2. Partitioned matrices A matrix A is a (2 x 2) block matrix if it is...
2. Partitioned matrices A matrix A is a (2 x 2) block matrix if it is represented in the form [ A 1 A2 1 A = | A3 A4 where each of the A; are matrices. Note that the matrix A need not be a square matrix; for instance, A might be (7 x 12) with Aj being (3 x 5), A2 being (3 x 7), A3 being (4 x 5), and A4 being (4 x 7). We can...
Prove that any m x n matrix A of rank k can be written as A...
Prove that any m x n matrix A of rank k can be written as A = {k=1 u;v] where {u1, ..., Uk} and {V1, ... , Vk} are linearly independent sets. By SVD, any mxn matrix A of rank k can be written as A = {k=10;U;v] where {u1,..., Uk} and {V1, ... , Vk} are orthonormal sets and 01 > 02 > ... >0k > 0. For this problem, prove without using SVD.
<Problem 2> Answer the following questions about the square matrix A of order 3: A= III....
<Problem 2> Answer the following questions about the square matrix A of order 3: A= III. The square matrix B of order 3 is diagonalizable and meets AB=BA. prove that any eigenvector p of A is also an eigenvector of B. IV. Find the square matrix B of order 3 that meets B2 = A, where B is diagonalizable and all eigenvalues of B are positive. V. The square matrix X of order 3 is diagonalizable and meets AX =...
Let AA be an n×nn×n matrix. Prove that if x⃗ x→ is an eigenvector of AA...
Let AA be an n×nn×n matrix. Prove that if x⃗ x→ is an eigenvector of AA corresponding to the eigenvalue λλ, then x⃗ x→ is also an eigenvector of A+cIA+cI, where cc is a scalar. Moreover, find the corresponding eigenvalue of A+cIA+cI.
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3]...
Advanced Calculus
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion.
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....
Here is the matrix mentioned in the problem:
7) Change (23 pts) Diagonalization of a matrix. For the matrix from the last problem. a) (6 pts) Find the eigenvalues of A b) (6 pts) Find the eigenvectors of A, Choose x1=1 for the first element of both eigenvectors. c) (4 pts) Is this system stable? What type of damping does it have? d) (2 pts) Find the diagonal matrix D corresponding to A. e) (1 pts) Find D3, use actual...
3. Suppose A is a real square n x n matrix with SVD given by A USVT Using MATLAB's eig and svd, investigate how the eigenvalues and eigen- vectors of the real symmetric matrix AT 0 depend on 2, U and V. Try a random matrix with n 2 to get started. Once you see the relationship, state it carefully, without proof. 4. (This is a continuation of the previous question.) Prove the property that you observed in the previous...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Question 3 (a) Use the defining equation for eigenvalues and eigenvectors to prove that if matrix A has the unique eigenvalue a, then the matrix A-al has 0 as an eigenvalue (b) Show that if matrix B has the eigenvalue then the matri B has 2 as an eigenvalue. (c) Use the defining equation to show that if the matrix C is invertible, then C cannot have zero as an eigenvalue. (Hint: No eigenvector Xcan be the zero vector. So...
2. Partitioned matrices A matrix A is a (2 x 2) block matrix if it is represented in the form [ A 1 A2 1 A = | A3 A4 where each of the A; are matrices. Note that the matrix A need not be a square matrix; for instance, A might be (7 x 12) with Aj being (3 x 5), A2 being (3 x 7), A3 being (4 x 5), and A4 being (4 x 7). We can...
Prove that any m x n matrix A of rank k can be written as A = {k=1 u;v] where {u1, ..., Uk} and {V1, ... , Vk} are linearly independent sets. By SVD, any mxn matrix A of rank k can be written as A = {k=10;U;v] where {u1,..., Uk} and {V1, ... , Vk} are orthonormal sets and 01 > 02 > ... >0k > 0. For this problem, prove without using SVD.
Advanced Calculus
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion.
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....
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