Homework Answers
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue...
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
Material: 8.3.2 Consider the matrix (1 2 3 A-2 3 1 (8.3.28) (i) Use (8.3.27) to find the dominant eigenvalue of A. (ii) Check to see that u-(1 , I , î ), is a positive eigenvector of A. Use 11 and T...
Material:
8.3.2 Consider the matrix (1 2 3 A-2 3 1 (8.3.28) (i) Use (8.3.27) to find the dominant eigenvalue of A. (ii) Check to see that u-(1 , I , î ), is a positive eigenvector of A. Use 11 and Theorem 8.6 to find the dominant eigenvalue of A and confirm that this is exactly what was obtained in part 0) obtained in part (i) or(ii ii) Compute all the eigenvalues of A directly and confirm the result...
The 2 x 2 matrix 1 = ( 43 II has two distinct real eigenvalues. 1....
The 2 x 2 matrix 1 = ( 43 II has two distinct real eigenvalues. 1. Give the characteristic polynomial for A in Maple notation in the form t^2 + a*t + b Characteristic polynomial = 2. Find the set of eigenvalues for A, enclosed in braces , ) with the two eigenvalues separated by a comma, like (-4, 7) Set of eigenvalues for A = 5 3. Find one eigenvector for each eigenvalue, using Maple > for vectors, e.g....
The matrix A= is diagonalisable with eigenvalues 1, -2 and -2. An eigenvector corresponding to the...
The matrix A= is diagonalisable with eigenvalues 1, -2 and -2.
An eigenvector corresponding to the eigenvalue 1 is . Find an
invertible matrix M such that M−1AM= ⎛⎝⎜⎜⎜1000-2000-2⎞⎠⎟⎟⎟. Enter
the Matrix M in the box below.
Question 8: Score 0/2 1 3 -3 4 6 -6 8 The matrix A = 1-6 6 | is diagonalisable with eigenvalues 1,-2 and-2. An eigenvector corresponding to the eigenvalue 1 is -2 2 1 0 0 0 0-2 Find an invertible matrix...
For the 3×2 matrix A: a) Determine the eigenvalues of ATA, and confirm that your eigenvalues...
For the 3×2 matrix A:
a) Determine the eigenvalues of ATA, and confirm that
your eigenvalues are consistent with the trace and determinant of
ATA.
b) Find an eigenvector for each eigenvalue of
ATA.
c) Find an invertible matrix P and a diagonal matrix D such that
P-1(ATA)P = D.
d) Find the singular value decomposition of the matrix A; that
is, find matrices U, Σ, and V such that A = UΣVT.
e) What is the best rank 1...
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3,...
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3, A2 = 2, 13 = 1 Find a basis B = {V1, V2, v3} for R3 consisting of eigenvectors of A. Give the corresponding eigenvalue for each eigenvector vi.
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...
Problem 5: Let A be the following matrix: 2 -3 1] A= 1 -2 11 1...
Problem 5: Let A be the following matrix: 2 -3 1] A= 1 -2 11 1 -3 2 (a) Compute the characteristic polynomial of A. (b) Find the eigenvalues of A. (c) For each eigenvalue of A, find a corresponding eigenvector.
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
Consider the following of the matrix A. Find all eigenvalues - 7,2 Give bases for each...
Consider the following of the matrix A. Find all eigenvalues - 7,2 Give bases for each of the corresponding eigenspaces smaller A-value spa larger A-value span and a diagonal matrix, such that 'AQ -0. (Enter each matrix in the form [row frow 2, ..., where each rows Orthogonally diagonalue the matrix by finding an orthogonal matrix comma-separated list) (0,0) -
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
Material:
8.3.2 Consider the matrix (1 2 3 A-2 3 1 (8.3.28) (i) Use (8.3.27) to find the dominant eigenvalue of A. (ii) Check to see that u-(1 , I , î ), is a positive eigenvector of A. Use 11 and Theorem 8.6 to find the dominant eigenvalue of A and confirm that this is exactly what was obtained in part 0) obtained in part (i) or(ii ii) Compute all the eigenvalues of A directly and confirm the result...
The 2 x 2 matrix 1 = ( 43 II has two distinct real eigenvalues. 1. Give the characteristic polynomial for A in Maple notation in the form t^2 + a*t + b Characteristic polynomial = 2. Find the set of eigenvalues for A, enclosed in braces , ) with the two eigenvalues separated by a comma, like (-4, 7) Set of eigenvalues for A = 5 3. Find one eigenvector for each eigenvalue, using Maple > for vectors, e.g....
The matrix A= is diagonalisable with eigenvalues 1, -2 and -2.
An eigenvector corresponding to the eigenvalue 1 is . Find an
invertible matrix M such that M−1AM= ⎛⎝⎜⎜⎜1000-2000-2⎞⎠⎟⎟⎟. Enter
the Matrix M in the box below.
Question 8: Score 0/2 1 3 -3 4 6 -6 8 The matrix A = 1-6 6 | is diagonalisable with eigenvalues 1,-2 and-2. An eigenvector corresponding to the eigenvalue 1 is -2 2 1 0 0 0 0-2 Find an invertible matrix...
For the 3×2 matrix A:
a) Determine the eigenvalues of ATA, and confirm that
your eigenvalues are consistent with the trace and determinant of
ATA.
b) Find an eigenvector for each eigenvalue of
ATA.
c) Find an invertible matrix P and a diagonal matrix D such that
P-1(ATA)P = D.
d) Find the singular value decomposition of the matrix A; that
is, find matrices U, Σ, and V such that A = UΣVT.
e) What is the best rank 1...
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3, A2 = 2, 13 = 1 Find a basis B = {V1, V2, v3} for R3 consisting of eigenvectors of A. Give the corresponding eigenvalue for each eigenvector vi.
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...
Problem 5: Let A be the following matrix: 2 -3 1] A= 1 -2 11 1 -3 2 (a) Compute the characteristic polynomial of A. (b) Find the eigenvalues of A. (c) For each eigenvalue of A, find a corresponding eigenvector.
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
Consider the following of the matrix A. Find all eigenvalues - 7,2 Give bases for each of the corresponding eigenspaces smaller A-value spa larger A-value span and a diagonal matrix, such that 'AQ -0. (Enter each matrix in the form [row frow 2, ..., where each rows Orthogonally diagonalue the matrix by finding an orthogonal matrix comma-separated list) (0,0) -
Most questions answered within 3 hours.
-
39.4% of US homes continue to use a landline in addition to cell
phone service. 3...
asked 34 minutes ago -
Starting with benzene, synthesize 1-phenyl-1-butyne.
Show intermediates and reagents.
asked 1 hour ago -
Create a 32-run crossed array design with six control factors
and two noise factors such that...
asked 2 hours ago -
A 500g sample of sand from source A has the following amounts
retained on each sieve....
asked 2 hours ago -
In
your own words, please explain the essay by John Keynes wrote "The
End of Laissez...
asked 2 hours ago -
How are the matrix and pixels related? Why are smaller
pixels better for diagnostic quality?
asked 2 hours ago -
2. An AC generator has 80 rectangular loops on
its armature. Each loop is 11 cm...
asked 2 hours ago -
Please help me with this question. Consider Aldi’s current and
potential geographic markets (see Exhibit 4...
asked 2 hours ago -
What are the main components of the fermentation process and
give an explanation of each? Include...
asked 2 hours ago -
Explain which types of cells in the body (belonging to which
organs, etc.) are sensitive to...
asked 2 hours ago -
A single cable supports an 703-kg elevator car. What is the
tension in the cable when...
asked 3 hours ago -
among the three different ways to link CSS specifications to an
HTML document (inline CSS, document...
asked 3 hours ago