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Linear Algebra
Suppose that B=P-1AP. (a) Prove that A and B have the same eigenvalues. (b)...
8: Suppose that A and B are similar matrices, B = p-1AP, where 6 2 P...
8: Suppose that A and B are similar matrices, B = p-1AP, where 6 2 P = -1 1 We know that A and B have the same characteristic polynomial and the same eigenvalues. Suppose that 2 is one of the common eigenvalues and x = [4 1] is a corresponding eigenvector of A. Which of the following is the eigenvector of B corresponding to 2 ? "f1-4-0-0-0-0-01-10 #8: Select
Please help with these linear algebra study guide so I can have correct answers to study...
Please help with these linear algebra study guide so I can have
correct answers to study with
5. Suppose that B = P-1AP. (a) Prove that A and B have the same eigenvalues. (b) Prove that if x is an eigenvector of A, then P-1x is an edigenvector of B. 6. Let A= 0.7 0.1 0.3 0.9 Find P and D such that A = PDP-1, where D is a diagonal matrix. 7. This is a continuation of the previous...
Linear Algebra 4. Prove that the eigenvalues of A and AT are identical. 5. Prove that...
Linear Algebra
4. Prove that the eigenvalues of A and AT are identical. 5. Prove that the eigenvalues of a diagonal matrix are equal to the diagonal elements. 6. Consider the matrix ompute the eigenvalues and eigenvectors of A, A-,
linear algebra 3. Suppose that A is a 2 x 2 matrix: (a) Find Az if...
linear algebra
3. Suppose that A is a 2 x 2 matrix: (a) Find Az if r = (13) is an eigenvector with eigenvalue 1 = 3. (b) Is it possible for A to have 3 eigenvalues? Why or why not? (C) True/False: If is an eigenvalue of A, there are infinitely many eigenvectors with eigenvalue .. (d) True/False: If I = 0) is an eigenvalue, then Eo = Nul (A).
linear algebra (1 point) Prove that if X+0 is an eigenvalue of an invertible matrix A,...
linear algebra
(1 point) Prove that if X+0 is an eigenvalue of an invertible matrix A, then is an eigenvalue of A! Proof: Suppose v is an eigenvector of eigenvalue then Au=du. Since A is invertible, we can multiply both sides of Au= du by 50 Az = Azj. This implies that . Since 1 + 0 we obtain that Thus – is an eigenvalue of A-? A.D=AU B. A=X co=A D. X-A7 = E. A- F. Av= < P...
Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that...
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
(a) Find the eigenvalues of the matrix 4) 2 1' and find an eigenvector corresponding to each eigenvalue. Hence find an invertible matrix, P, and a diagonal matrix, D, such that P-1AP = D. (b) Use...
(a) Find the eigenvalues of the matrix 4) 2 1' and find an eigenvector corresponding to each eigenvalue. Hence find an invertible matrix, P, and a diagonal matrix, D, such that P-1AP = D. (b) Use your result from (a) to find the functions f(t) and g(t) such that f(t)-f(t) +2g(t) g(t) 2f(t) g(t), where f(0)-1 and g(0)-2 (c) Now suppose that f(0)-α and g(0) β. Determine the condition(s) on α and β that must hold if, as t,t is...
linear algebra Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are...
linear algebra
Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal -1 0-1 0-1 0 - 1 0 9 1 Find the characteristic polynomial of A. |x - Al- Find the eigenvalues of A. (Enter your answers from smallest to largest.) (21, 22, 23) Find the general form for every eigenvector corresponding to 21. (Uses as your parameter.) X1 - Find the general form for every elge vector corresponding to Az. (Uset as your...
Linear algebra, please have a legible answer. Thank you. 6. [10 points) Find a matrix that...
Linear algebra, please have a legible answer. Thank
you.
6. [10 points) Find a matrix that diagonalizes A and determine P-1AP.
linear algebra Use the matrix P to determine if the matrices A and A' are similar....
linear algebra
Use the matrix P to determine if the matrices A and A' are similar. P = 15 9 -20 -11 1 p-1 p-1AP = Are they similar? Yes, they are similar. No, they are not similar.
8: Suppose that A and B are similar matrices, B = p-1AP, where 6 2 P = -1 1 We know that A and B have the same characteristic polynomial and the same eigenvalues. Suppose that 2 is one of the common eigenvalues and x = [4 1] is a corresponding eigenvector of A. Which of the following is the eigenvector of B corresponding to 2 ? "f1-4-0-0-0-0-01-10 #8: Select
Please help with these linear algebra study guide so I can have
correct answers to study with
5. Suppose that B = P-1AP. (a) Prove that A and B have the same eigenvalues. (b) Prove that if x is an eigenvector of A, then P-1x is an edigenvector of B. 6. Let A= 0.7 0.1 0.3 0.9 Find P and D such that A = PDP-1, where D is a diagonal matrix. 7. This is a continuation of the previous...
Linear Algebra
4. Prove that the eigenvalues of A and AT are identical. 5. Prove that the eigenvalues of a diagonal matrix are equal to the diagonal elements. 6. Consider the matrix ompute the eigenvalues and eigenvectors of A, A-,
linear algebra
3. Suppose that A is a 2 x 2 matrix: (a) Find Az if r = (13) is an eigenvector with eigenvalue 1 = 3. (b) Is it possible for A to have 3 eigenvalues? Why or why not? (C) True/False: If is an eigenvalue of A, there are infinitely many eigenvectors with eigenvalue .. (d) True/False: If I = 0) is an eigenvalue, then Eo = Nul (A).
linear algebra
(1 point) Prove that if X+0 is an eigenvalue of an invertible matrix A, then is an eigenvalue of A! Proof: Suppose v is an eigenvector of eigenvalue then Au=du. Since A is invertible, we can multiply both sides of Au= du by 50 Az = Azj. This implies that . Since 1 + 0 we obtain that Thus – is an eigenvalue of A-? A.D=AU B. A=X co=A D. X-A7 = E. A- F. Av= < P...
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
(a) Find the eigenvalues of the matrix 4) 2 1' and find an eigenvector corresponding to each eigenvalue. Hence find an invertible matrix, P, and a diagonal matrix, D, such that P-1AP = D. (b) Use your result from (a) to find the functions f(t) and g(t) such that f(t)-f(t) +2g(t) g(t) 2f(t) g(t), where f(0)-1 and g(0)-2 (c) Now suppose that f(0)-α and g(0) β. Determine the condition(s) on α and β that must hold if, as t,t is...
linear algebra
Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal -1 0-1 0-1 0 - 1 0 9 1 Find the characteristic polynomial of A. |x - Al- Find the eigenvalues of A. (Enter your answers from smallest to largest.) (21, 22, 23) Find the general form for every eigenvector corresponding to 21. (Uses as your parameter.) X1 - Find the general form for every elge vector corresponding to Az. (Uset as your...
Linear algebra, please have a legible answer. Thank
you.
6. [10 points) Find a matrix that diagonalizes A and determine P-1AP.
linear algebra
Use the matrix P to determine if the matrices A and A' are similar. P = 15 9 -20 -11 1 p-1 p-1AP = Are they similar? Yes, they are similar. No, they are not similar.
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