(c) Use the results of parts (a) and (b) to write down all eigenvalues of A along with their algebraic and geometric multiplicities. Is A diagonalizable? (Note: This question does not require finding eigenvalues by solving det(A XI) 0)
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5. Let -2 0 2AA8 (a) Show thatis an eigenvector of A. What is its eigenvalue? (b) By solving (A+2/)x 0, show that -2 is an eigenvalue of A. (c) Use the results of parts (a) and (b) to write down all...
Question 4: Eigenvalue Theory 2 Let A Cnxn. For each of the following statements show that it is ...
Question 4: Eigenvalue Theory 2 Let A Cnxn. For each of the following statements show that it is true or give a counterexample to show that it is false (a) If λ is an eigenvalue of A, and μ є Cn then λ-μ is an eigenvalue of A-1 (b) If A is real and λ is an eigenvalue of A then so is-λ. (c) If A is real and λ is an eigenvalue of A, then so is λ. (d)...
Let A be the matrix To 1 0] A= -4 4 0 1-2 0 1 (a)...
Let A be the matrix To 1 0] A= -4 4 0 1-2 0 1 (a) Find the eigenvalues and eigenvectors of A. (b) Find the algebraic multiplicity an, and the geometric multiplicity, g, of each eigenvalue. (c) For one of the eigenvalues you should have gi < az. (If not, redo the preceding parts!) Find a generalized eigenvector for this eigenvalue. (d) Verify that the eigenvectors and generalized eigenvectors are all linearly independent. (e) Find a fundamental set of...
0 2 0 Q1) Let A = 1 3 2 2 0 a) Determine all eigenvalues...
0 2 0 Q1) Let A = 1 3 2 2 0 a) Determine all eigenvalues of A. b) Determine the basis for each eigenspace of A c) Determine the algebraic and geometric multiplicity of each eigenvalue. Q2) Let aj, 02, 03, 04, agbe real numbers. Compute ai det 1 1 Q3) Determine all values of x E R such that matrix 4 0 3 х 2 5 A = is invertable. х 0 0 1 0 0 4 0
4. (a) Write down, without proof, all parts of the Perron-Frobenius Theorem (b) Let S be a stochastic matrix. Prove that 1 is the Perron eigenvalue of S, and e (1 Furthermore, prove that A-1 for ever...
4. (a) Write down, without proof, all parts of the Perron-Frobenius Theorem (b) Let S be a stochastic matrix. Prove that 1 is the Perron eigenvalue of S, and e (1 Furthermore, prove that A-1 for every eigenvalue λ of S 1) is the corresponding Perron eigenvector of S (c) For each of the given matrices S(a) below determine the values of the parameter di for which the limit link oo (Si exists. Justify your answer! 1 1 2 2...
hi, can you write the notation and everything? Assignment 12 Let A= -3 -5 -2 6...
hi, can you write the notation
and everything?
Assignment 12 Let A= -3 -5 -2 6 8 2 0 0 -2 [3pt] (a) Find the eigenvalues of A. (1pt] (b) Determine the algebraic multiplicity of each eigenvalue of A. [3pt] (c) Find a basis for each eigenspace for A. (1pt] (d) Determine the geometric multiplicity of each eigenvalue of A. [2pt] (e) Give a matrix P and a diagonal matrix D such that P-1AP = D.
3. Let L be the linear transformation on R2 that reflects each point P across the line y kx (k> 0) are eigenvectors of L a) (2 marks) Show that v1 and vz b) (1 mark) What is the eigenvalue corresp...
3. Let L be the linear transformation on R2 that reflects each point P across the line y kx (k> 0) are eigenvectors of L a) (2 marks) Show that v1 and vz b) (1 mark) What is the eigenvalue corresponding to each eigenvector? (Hint: No need to calculate the characteristic polynomial or solve a matrix equation. Geometric reasoning should suffice to solve this problem. Drawing a diagram is recommended!)
3. Let L be the linear transformation on R2 that...
4 7 5 0 2 2 Problem 7 Let A= -1 2 9 -4 1 5...
4 7 5 0 2 2 Problem 7 Let A= -1 2 9 -4 1 5 -1 3 7 3 1 -4 2 0 1 1 0 10 2 a) (4 pts] Using the [V, D] command in MATLAB with rational format, find a diagonal matrix D and a matrix V of maximal rank satisfying the matrix equation A * V = V * D. Is A real-diagonalizable? b) [4 pts) Write down the eigenvalues of A. For each eigenvalue,...
1 Problem 7 Let A 4 5 - 1 5 0 2 -1 2 3 -4...
1 Problem 7 Let A 4 5 - 1 5 0 2 -1 2 3 -4 7 2 1 3 7 2 -4 2 0 0 10 1 1 a) (4 pts] Using the [V, DJ command in MATLAB with rational format, find a diagonal matrix D and a matrix V of maximal rank satisfying the matrix equation A * V = V * D. Is A real-diagonalizable? b) (4 pts) Write down the eigenvalues of A. For each eigenvalue,...
Let A = 2 −5 7 4 . The vector −1 − √ 34i 7 is an eigenvector for A with eigenvalue 3 − √ 34i. Find the other eigenvalue and a vector in its eigenspace Answers: a) 3 + √ 34i, −1 − √ 34i 7 b) 3 − √ 34i, 1 + √ 34i 7 c) 3 + √ 34i, −1 + √ 34i 7 see
Let A = 2 −5 7 4 . The vector −1 − √ 34i 7 is an eigenvector for A with eigenvalue 3 − √ 34i. Find the other eigenvalue and a vector in its eigenspace Answers: a) 3 + √ 34i, −1 − √ 34i 7 b) 3 − √ 34i, 1 + √ 34i 7 c) 3 + √ 34i, −1 + √ 34i 7see photo
5. Let (a) (2 marks) Find all eigenvalues of A (b) (4 marks) Find an orthonormal basis for each e...
5. Let (a) (2 marks) Find all eigenvalues of A (b) (4 marks) Find an orthonormal basis for each eigenspace of A (you may find an orthonormal basis by inspection or use the Gram-Schmidt algorithm on each eigenspace) (c) (2 marks) Deduce that A is orthogonally diagonalizable. Write down an orthogonal matrix P and a diagonal matrix D such that D P-AP. (d) (1 mark) Use the fact that P is an orthogonal matrix to find P-1 (e) (2 marks)...
Question 4: Eigenvalue Theory 2 Let A Cnxn. For each of the following statements show that it is true or give a counterexample to show that it is false (a) If λ is an eigenvalue of A, and μ є Cn then λ-μ is an eigenvalue of A-1 (b) If A is real and λ is an eigenvalue of A then so is-λ. (c) If A is real and λ is an eigenvalue of A, then so is λ. (d)...
Let A be the matrix To 1 0] A= -4 4 0 1-2 0 1 (a) Find the eigenvalues and eigenvectors of A. (b) Find the algebraic multiplicity an, and the geometric multiplicity, g, of each eigenvalue. (c) For one of the eigenvalues you should have gi < az. (If not, redo the preceding parts!) Find a generalized eigenvector for this eigenvalue. (d) Verify that the eigenvectors and generalized eigenvectors are all linearly independent. (e) Find a fundamental set of...
0 2 0 Q1) Let A = 1 3 2 2 0 a) Determine all eigenvalues of A. b) Determine the basis for each eigenspace of A c) Determine the algebraic and geometric multiplicity of each eigenvalue. Q2) Let aj, 02, 03, 04, agbe real numbers. Compute ai det 1 1 Q3) Determine all values of x E R such that matrix 4 0 3 х 2 5 A = is invertable. х 0 0 1 0 0 4 0
4. (a) Write down, without proof, all parts of the Perron-Frobenius Theorem (b) Let S be a stochastic matrix. Prove that 1 is the Perron eigenvalue of S, and e (1 Furthermore, prove that A-1 for every eigenvalue λ of S 1) is the corresponding Perron eigenvector of S (c) For each of the given matrices S(a) below determine the values of the parameter di for which the limit link oo (Si exists. Justify your answer! 1 1 2 2...
hi, can you write the notation
and everything?
Assignment 12 Let A= -3 -5 -2 6 8 2 0 0 -2 [3pt] (a) Find the eigenvalues of A. (1pt] (b) Determine the algebraic multiplicity of each eigenvalue of A. [3pt] (c) Find a basis for each eigenspace for A. (1pt] (d) Determine the geometric multiplicity of each eigenvalue of A. [2pt] (e) Give a matrix P and a diagonal matrix D such that P-1AP = D.
3. Let L be the linear transformation on R2 that reflects each point P across the line y kx (k> 0) are eigenvectors of L a) (2 marks) Show that v1 and vz b) (1 mark) What is the eigenvalue corresponding to each eigenvector? (Hint: No need to calculate the characteristic polynomial or solve a matrix equation. Geometric reasoning should suffice to solve this problem. Drawing a diagram is recommended!)
3. Let L be the linear transformation on R2 that...
4 7 5 0 2 2 Problem 7 Let A= -1 2 9 -4 1 5 -1 3 7 3 1 -4 2 0 1 1 0 10 2 a) (4 pts] Using the [V, D] command in MATLAB with rational format, find a diagonal matrix D and a matrix V of maximal rank satisfying the matrix equation A * V = V * D. Is A real-diagonalizable? b) [4 pts) Write down the eigenvalues of A. For each eigenvalue,...
1 Problem 7 Let A 4 5 - 1 5 0 2 -1 2 3 -4 7 2 1 3 7 2 -4 2 0 0 10 1 1 a) (4 pts] Using the [V, DJ command in MATLAB with rational format, find a diagonal matrix D and a matrix V of maximal rank satisfying the matrix equation A * V = V * D. Is A real-diagonalizable? b) (4 pts) Write down the eigenvalues of A. For each eigenvalue,...
5. Let (a) (2 marks) Find all eigenvalues of A (b) (4 marks) Find an orthonormal basis for each eigenspace of A (you may find an orthonormal basis by inspection or use the Gram-Schmidt algorithm on each eigenspace) (c) (2 marks) Deduce that A is orthogonally diagonalizable. Write down an orthogonal matrix P and a diagonal matrix D such that D P-AP. (d) (1 mark) Use the fact that P is an orthogonal matrix to find P-1 (e) (2 marks)...
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