492) Calculator exercise. For the matrix [1 7; 4 2] determine the ordered eigenvalues (e1,e2), where...
Homework Answers
For matrix 
, eigenvalue 
and eigenvector v is defined by
eigenvalues satisfy the equation, ,
The roots are eigenvalues.
The eigen vectors are given by For e1
.
Now, we have to reduce the matrix by row operations.
Perform 
So, for [1;v1], 
The eigen vectors are given by For e1
Perform 
So, for [1;v2], 
The required eigenvectors are:
,
and 
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492) Calculator exercise. For the matrix [1 7; 4 2] determine the
ordered eigenvalues (e1,e2), where...
(4 points) Given that the matrix A has eigenvalues 21 = -7 with corresponding eigenvector vi...
(4 points) Given that the matrix A has eigenvalues 21 = -7 with corresponding eigenvector vi = N and 22 = 3 with corresponding eigenvector v2 = , find A. A=
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.
(1 point) Find the eigenvalues of the matrix A . -19 6 0 0 -36 11...
(1 point) Find the eigenvalues of the matrix A . -19 6 0 0 -36 11 0 0 A= The eigenvalues are λ| < λ2 < λ3 < λ4, where has an eigenvector 12 has an eigenvector has an eigenvector 4 has an eigenvector Note: you may want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues
3. Find all the eigenvalues and corresponding eigenspaces for the matrix B = 4. Show that...
3. Find all the eigenvalues and corresponding eigenspaces for the matrix B = 4. Show that the matrix B = 0 1 is not diagonalizable. 0 4] Lo 5. Let 2, and 1, be two distinct eigenvalues of a matrix A (2, # 12). Assume V1, V2 are eigenvectors of A corresponding to 11 and 22 respectively. Prove that V1, V2 are linearly independent.
Question 2 (1 point) 8 -18 Find the eigenvalues and eigenvectors of the matrix A =...
Question 2 (1 point) 8 -18 Find the eigenvalues and eigenvectors of the matrix A = 18] (The 3 -7 same as in the previous problem.) di = 2, V1 = [1] and 12 = -1, V2 = - [11] [1] 3 21 = 1, V1 = ܒܗ ܟܬ and 12 = -2, V2 = 2 x = 1, V1 = and 12 = -2, V2 = [11 11 x = -2, Vi and 12 = -3, V2 [1]
(1 point) Given that ū = and are eigenvectors of the matrix -12 24 determine the...
(1 point) Given that ū = and are eigenvectors of the matrix -12 24 determine the corresponding eigenvalues. 21 = -1 12 = 1 (1 point) Solve the system -6 1 dx dt х -6 -1 with the initial value 0 x(0) = -2 x(t) = (1 point) Calculate the eigenvalues of this matrix: [Note-- you'll probably want to use a calculator or computer to estimate the roots of the polynomial which defines the eigenvalues. You also may want to...
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...
In Exercise, find the eigenvalues of each matrix and determine a basis for each eigenspace -7...
In Exercise, find the eigenvalues of each matrix and determine a basis for each eigenspace -7 5 4 0 -3 -8 9 5
Find all eigenvalues and eigenvector of the matrix 2 2 A 1 1 -2 -4-1 Give...
Find all eigenvalues and eigenvector of the matrix 2 2 A 1 1 -2 -4-1 Give the eigenvalues in ascending order. Choose the corresponding eigenvectors from the table below: 0 1 -2 2 1 V 2 = A 0 2 Vector 1 Vector 2 Vector 3 Vector 4 Vector 5 Vector 6 Eigenvector number: Eigenvector number: A3 Eigenvector number: Il
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...
(4 points) Given that the matrix A has eigenvalues 21 = -7 with corresponding eigenvector vi = N and 22 = 3 with corresponding eigenvector v2 = , find A. A=
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.
(1 point) Find the eigenvalues of the matrix A . -19 6 0 0 -36 11 0 0 A= The eigenvalues are λ| < λ2 < λ3 < λ4, where has an eigenvector 12 has an eigenvector has an eigenvector 4 has an eigenvector Note: you may want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues
3. Find all the eigenvalues and corresponding eigenspaces for the matrix B = 4. Show that the matrix B = 0 1 is not diagonalizable. 0 4] Lo 5. Let 2, and 1, be two distinct eigenvalues of a matrix A (2, # 12). Assume V1, V2 are eigenvectors of A corresponding to 11 and 22 respectively. Prove that V1, V2 are linearly independent.
Question 2 (1 point) 8 -18 Find the eigenvalues and eigenvectors of the matrix A = 18] (The 3 -7 same as in the previous problem.) di = 2, V1 = [1] and 12 = -1, V2 = - [11] [1] 3 21 = 1, V1 = ܒܗ ܟܬ and 12 = -2, V2 = 2 x = 1, V1 = and 12 = -2, V2 = [11 11 x = -2, Vi and 12 = -3, V2 [1]
(1 point) Given that ū = and are eigenvectors of the matrix -12 24 determine the corresponding eigenvalues. 21 = -1 12 = 1 (1 point) Solve the system -6 1 dx dt х -6 -1 with the initial value 0 x(0) = -2 x(t) = (1 point) Calculate the eigenvalues of this matrix: [Note-- you'll probably want to use a calculator or computer to estimate the roots of the polynomial which defines the eigenvalues. You also may want to...
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...
In Exercise, find the eigenvalues of each matrix and determine a basis for each eigenspace -7 5 4 0 -3 -8 9 5
Find all eigenvalues and eigenvector of the matrix 2 2 A 1 1 -2 -4-1 Give the eigenvalues in ascending order. Choose the corresponding eigenvectors from the table below: 0 1 -2 2 1 V 2 = A 0 2 Vector 1 Vector 2 Vector 3 Vector 4 Vector 5 Vector 6 Eigenvector number: Eigenvector number: A3 Eigenvector number: Il
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...
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