Homework Answers
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)...
Consider the 3 x 3 matrix A defined as follows 7 4-4 a) Find the eigenvalues...
Consider the 3 x 3 matrix A defined as follows 7 4-4 a) Find the eigenvalues of A. Is A singular matrix? b) Find a basis for each eigenspace. Then, determine their dimensions c) Find the eigenvalues of A10 and their corresponding eigenspaces. d) Do the eigenvectors of A form a basis for IR3? e) Find an orthogonal matrix P that diagonalizes A f) Use diagonalization to compute A 6
The eigenvalues of the symmetric matrix A= ſi 8 41 8 1 -4 are 11 =...
The eigenvalues of the symmetric matrix A= ſi 8 41 8 1 -4 are 11 = 9 and 12 = -9. 14 -4 7 | Find an orthogonal diagonalization of A. Find the characteristic polynomial of A.
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let P2 R2 R2 be the orthogonal projection on...
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let P2 R2 R2 be the orthogonal projection onto the line L2: x32 2r2 0. 0. (1) What are the image and kernel of P2P What is the rank of P2P? Give a geometric description, without relying on the matrix of P2P (2) Find the matrix that represents P2P
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let...
11 18 7 Let 4 6 10 (a) Find the eigenvalues of A. (b) For each...
11 18 7 Let 4 6 10 (a) Find the eigenvalues of A. (b) For each eigenvalue find the corresponding eigenvectors. (c) Let 21 and 22 be the eigenvalues of A such that 21 <12. Find a match for 11. Find a match for 12 Find a matching eigenvector vị for 11 - Find a matching eigenvector v2 for 12 Let P and D be 2 x 2 matrices defined as follows: 20 and P = [v1v2] o 22 that...
Let A = -2 -2 6 1 -2. -2 co (a) Compute eigenvalues and corresponding eigenvectors...
Let A = -2 -2 6 1 -2. -2 co (a) Compute eigenvalues and corresponding eigenvectors of A. (b) Find an invertible matrix P such that P-1AP is diagonal. (c) Find an orthogonal matrix Q (that is QT = Q-1 ) such that QTAQ is diagonal (d) Compute e At
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...
Corresponding eigenvectors of each eigenvalue 9 Let 2. (as find the eigenvalues of A GA 1...
Corresponding eigenvectors of each eigenvalue 9 Let 2. (as find the eigenvalues of A GA 1 -- 1 and find the or A each 5 Find the corresponding eigenspace to each eigen value of A. Moreover, Find a basis for The Corresponding eigenspace (c) Determine whether A is diagonalizable. If it is, Find a diagonal matrix ) and an invertible matrix P such that p-AP=1
) Let A be the following matrix: 13 0 2 0 2 2 0 0 6...
) Let A be the following matrix: 13 0 2 0 2 2 0 0 6 (a) Enter its characteristic equation below. Note you must use p as the parameter instead of , and you must enter your answer as a equation, with the equals sign. (b) Enter the eigenvalues of the matrix, including any repetition. For example 16,16,24. 5 (c) Find the eigenvectors, and then use Gram-Schmidt to find an orthonormal basis for each eigenvalue's eigenspace. Build an orthogonal...
Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal. -1...
Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal. -1 0 -1 0-1 0 - 1 0 5 Find the characteristic polynomial of A. - A - Find the eigenvalues of A. (Enter your answers from smallest to largest.) (1, 12, 13) = ]) Find the general form for every eigenvector corresponding to N. (Uses as your parameter.) X1 = Find the general form for every eigenvector corresponding to 12. (Use t as your...
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)...
Consider the 3 x 3 matrix A defined as follows 7 4-4 a) Find the eigenvalues of A. Is A singular matrix? b) Find a basis for each eigenspace. Then, determine their dimensions c) Find the eigenvalues of A10 and their corresponding eigenspaces. d) Do the eigenvectors of A form a basis for IR3? e) Find an orthogonal matrix P that diagonalizes A f) Use diagonalization to compute A 6
The eigenvalues of the symmetric matrix A= ſi 8 41 8 1 -4 are 11 = 9 and 12 = -9. 14 -4 7 | Find an orthogonal diagonalization of A. Find the characteristic polynomial of A.
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let P2 R2 R2 be the orthogonal projection onto the line L2: x32 2r2 0. 0. (1) What are the image and kernel of P2P What is the rank of P2P? Give a geometric description, without relying on the matrix of P2P (2) Find the matrix that represents P2P
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let...
11 18 7 Let 4 6 10 (a) Find the eigenvalues of A. (b) For each eigenvalue find the corresponding eigenvectors. (c) Let 21 and 22 be the eigenvalues of A such that 21 <12. Find a match for 11. Find a match for 12 Find a matching eigenvector vị for 11 - Find a matching eigenvector v2 for 12 Let P and D be 2 x 2 matrices defined as follows: 20 and P = [v1v2] o 22 that...
Let A = -2 -2 6 1 -2. -2 co (a) Compute eigenvalues and corresponding eigenvectors of A. (b) Find an invertible matrix P such that P-1AP is diagonal. (c) Find an orthogonal matrix Q (that is QT = Q-1 ) such that QTAQ is diagonal (d) Compute e At
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...
Corresponding eigenvectors of each eigenvalue 9 Let 2. (as find the eigenvalues of A GA 1 -- 1 and find the or A each 5 Find the corresponding eigenspace to each eigen value of A. Moreover, Find a basis for The Corresponding eigenspace (c) Determine whether A is diagonalizable. If it is, Find a diagonal matrix ) and an invertible matrix P such that p-AP=1
) Let A be the following matrix: 13 0 2 0 2 2 0 0 6 (a) Enter its characteristic equation below. Note you must use p as the parameter instead of , and you must enter your answer as a equation, with the equals sign. (b) Enter the eigenvalues of the matrix, including any repetition. For example 16,16,24. 5 (c) Find the eigenvectors, and then use Gram-Schmidt to find an orthonormal basis for each eigenvalue's eigenspace. Build an orthogonal...
Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal. -1 0 -1 0-1 0 - 1 0 5 Find the characteristic polynomial of A. - A - Find the eigenvalues of A. (Enter your answers from smallest to largest.) (1, 12, 13) = ]) Find the general form for every eigenvector corresponding to N. (Uses as your parameter.) X1 = Find the general form for every eigenvector corresponding to 12. (Use t as your...
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