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![Lets check what Eigen value comes for different n Let n=2 so A= ( 2 į] (A-21] = [?? 221 0 = (1-2) (2-2)-1 o=2+2²-32-2 o=7²-3](http://img.homeworklib.com/questions/c3b26060-c7b9-11eb-bc67-e1bba0513db5.png?x-oss-process=image/resize,w_560)
5. Consider the matrix A-1-6-7-3 Hint: The characteristic polynomial of A is p(λ ) =-(-2)0+ 1)2....
5. Consider the matrix A-1-6-7-3 Hint: The characteristic polynomial of A is p(λ ) =-(-2)0+ 1)2. (a) Find the eigenvalues of A and bases for the corresponding eigenspaces. (b) Determine the geometric and algebraic multiplicities of each eigenvalue and whether A is diagonalizable or not. If it is, give a diagonal matrix D and an invertible matrix S such that A-SDS-1. If it's not, say why not.
2. (6 points). Consider a state space system: C1 =22 22 = - 2.c 1 -...
2. (6 points). Consider a state space system: C1 =22 22 = - 2.c 1 - 3.02 y=21 +22 Eco with Xo = (-1,1). (a) Specify the state space matrices (A,B,C,D). (b) Compute the matrix exponential eAl using similarity transformation. (e) Find the complete state response (solution of the SS system x(t)) if u(t) = 1. (d) Find the output response y(t) = Cx(t).
Review 4: question 1 Let A be an n x n matrix. Which of the below...
Review 4: question 1 Let A be an n x n matrix. Which of the below is not true? A. A scalar 2 is an eigenvalue of A if and only if (A - 11) is not invertible. B. A non-zero vector x is an eigenvector corresponding to an eigenvalue if and only if x is a solution of the matrix equation (A-11)x= 0. C. To find all eigenvalues of A, we solve the characteristic equation det(A-2) = 0. D)....
(1 point) The linear transformation T: R4 R4 below is diagonalizable. T(x,y,z,w) = (x – -...
(1 point) The linear transformation T: R4 R4 below is diagonalizable. T(x,y,z,w) = (x – - (2x + y), -z, 2 – 3w Compute the following. (Click to open and close sections below). (A) Characteristic Polynomial Compute the characteristic polynomial (as a function of t). A(t) = (B) Roots and Multiplicities Find the roots of A(t) and their algebraic multiplicities. Root Multiplicity t= t= t= t= (Leave any unneeded answer spaces blank.) (C) Eigenvalues and Eigenspaces Find the eigenvalues and...
1 2 Suppose that the non-singular n × n matrix A can be diagonalized, ie A...
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Suppose that the non-singular n × n matrix A can be diagonalized, ie A = PDP-1 where D is a diagonal matrix. Show that A-1 and AT can be diagonalized. 1.e. Suppose we have 2nu x 2n block matrices Y=I-I B O AB where all sub-matrices are n × n and O denotes the zero matrix. Find a block matrix X such that XY the determinant of X? Z and demonstrate it works. What is
-10 0] (1 point) Find a non-zero 2 x 2 matrix A such that A2
-10 0] (1 point) Find a non-zero 2 x 2 matrix A such that A2
Consider the matrix 3 -2 1 A 1 2 -1 1-2 3 a) Find the characteristic...
Consider the matrix 3 -2 1 A 1 2 -1 1-2 3 a) Find the characteristic polynomial of A and show that A has an eigenvalue at zero. Find the other two eigenvalues of A b) Find eigenvectors of A corresponding to all eigenvalues c) Can you diagonalize this matrix?
6. Let T P2 P be a linear transformation such that T P2P2 is still a...
6. Let T P2 P be a linear transformation such that T P2P2 is still a linear trans formation such that T(1) 2r22 T(2-)=2 T(1) = 2r22 T(12 - )=2 T(x2x= 2r T(r2)2x (a) (6 points) Find the matrix for T in some basis B. Specify the basis that you use. (d) (4 points) Find a basis for the eigenspace E2. (b) (2 points) Find det(T) and tr(T') (e) (4 points) Find a basis = (f,9,h) for P2 such that...
(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...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form,...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...
5. Consider the matrix A-1-6-7-3 Hint: The characteristic polynomial of A is p(λ ) =-(-2)0+ 1)2. (a) Find the eigenvalues of A and bases for the corresponding eigenspaces. (b) Determine the geometric and algebraic multiplicities of each eigenvalue and whether A is diagonalizable or not. If it is, give a diagonal matrix D and an invertible matrix S such that A-SDS-1. If it's not, say why not.
2. (6 points). Consider a state space system: C1 =22 22 = - 2.c 1 - 3.02 y=21 +22 Eco with Xo = (-1,1). (a) Specify the state space matrices (A,B,C,D). (b) Compute the matrix exponential eAl using similarity transformation. (e) Find the complete state response (solution of the SS system x(t)) if u(t) = 1. (d) Find the output response y(t) = Cx(t).
Review 4: question 1 Let A be an n x n matrix. Which of the below is not true? A. A scalar 2 is an eigenvalue of A if and only if (A - 11) is not invertible. B. A non-zero vector x is an eigenvector corresponding to an eigenvalue if and only if x is a solution of the matrix equation (A-11)x= 0. C. To find all eigenvalues of A, we solve the characteristic equation det(A-2) = 0. D)....
(1 point) The linear transformation T: R4 R4 below is diagonalizable. T(x,y,z,w) = (x – - (2x + y), -z, 2 – 3w Compute the following. (Click to open and close sections below). (A) Characteristic Polynomial Compute the characteristic polynomial (as a function of t). A(t) = (B) Roots and Multiplicities Find the roots of A(t) and their algebraic multiplicities. Root Multiplicity t= t= t= t= (Leave any unneeded answer spaces blank.) (C) Eigenvalues and Eigenspaces Find the eigenvalues and...
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Suppose that the non-singular n × n matrix A can be diagonalized, ie A = PDP-1 where D is a diagonal matrix. Show that A-1 and AT can be diagonalized. 1.e. Suppose we have 2nu x 2n block matrices Y=I-I B O AB where all sub-matrices are n × n and O denotes the zero matrix. Find a block matrix X such that XY the determinant of X? Z and demonstrate it works. What is
-10 0] (1 point) Find a non-zero 2 x 2 matrix A such that A2
Consider the matrix 3 -2 1 A 1 2 -1 1-2 3 a) Find the characteristic polynomial of A and show that A has an eigenvalue at zero. Find the other two eigenvalues of A b) Find eigenvectors of A corresponding to all eigenvalues c) Can you diagonalize this matrix?
6. Let T P2 P be a linear transformation such that T P2P2 is still a linear trans formation such that T(1) 2r22 T(2-)=2 T(1) = 2r22 T(12 - )=2 T(x2x= 2r T(r2)2x (a) (6 points) Find the matrix for T in some basis B. Specify the basis that you use. (d) (4 points) Find a basis for the eigenspace E2. (b) (2 points) Find det(T) and tr(T') (e) (4 points) Find a basis = (f,9,h) for P2 such that...
(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...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...
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