3 Eigenvalues and Eigenvectors Represent the matrix A as A = V-¹AV, where A = diag(A₁..... An) is the diagonal matrix with the eigenvalues A E C of A on its diagonal, and V = [...] is the matrix whose columns are the eigenvectors, such that 1 E C is an eigenvector associated with the eigen- value A. Whenever possible, normalize the eigenvector so that the first coordinate of the vector is 1 (if the first coordinate is 0, normalize it so that the second coordinate is 1). -5 3 (a) A= 18 (b) A= 1 Calculate A² and A²-3A +27. Calculate trace(A) and det(A). -6 [2 -0.5] (c) A= (Can you conclude what is A¹ without any additional calculation?) 3 -0.5
3 Eigenvalues and Eigenvectors Represent the matrix A as A = V-¹AV, where A = diag(A₁..... An) is the diagonal matrix with the eigenvalues A E C of A on its diagonal, and V = [...] is the matrix whose columns are the eigenvectors, such that 1 E C is an eigenvector associated with the eigen- value A. Whenever possible, normalize the eigenvector so that the first coordinate of the vector is 1 (if the first coordinate is 0, normalize it so that the second coordinate is 1). -5 3 (a) A= 18 (b) A= 1 Calculate A² and A²-3A +27. Calculate trace(A) and det(A). -6 [2 -0.5] (c) A= (Can you conclude what is A¹ without any additional calculation?) 3 -0.5
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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part A B C
![3 Eigenvalues and Eigenvectors
Represent the matrix A as A = V-¹AV, where A = diag(A... An) is the diagonal matrix
with the eigenvalues X EC of A on its diagonal, and V = [...] is the matrix whose
columns are the eigenvectors, such that , E C is an eigenvector associated with the eigen-
value A. Whenever possible, normalize the eigenvector so that the first coordinate of the
vector is 1 (if the first coordinate is 0, normalize it so that the second coordinate is 1).
(a) A=
-5 3
-18 10
(b) A =
1.
Calculate A² and A²-3A +27. Calculate trace(A) and det(A).
(c) A =
(Can you conclude what is A-¹ without any additional calculation?)
3 -0.5
(d) A=
(e) A =
10
-24 10
10] Calculate A² and A²-41. Calculate trace(A) and det(A).
.
(f) A =
-41-
... mind the complex values. Calculate A²-2A +21. Calculate trace(4)
and det(A).
(8) 4 = 41¹]. (This matrix represents contra-clockwise rotation by 45 degrees.)
If you like extra practice, you can find the decomposition using the following trick:
First calculate A², this is the matrix representing the contra-clockwise rotation by 90
degrees. Represent A² as V-¹AV, and then A = V-¹AV, where A solves the equation
A² = A (simply take square root of each entry on the diagonal, however, keep in mind
that both √ and -√ solve the equation X² = Ã; remember that Vi= (1 + i) and
i=(1-i)). You should then verify that indeed A = V-¹AV, because in fact there
are 4 matrices A such that A² = Ã. Note that this trick is not universal, for example
AS = I, but you won't be able to find the decomposition of A using the decomposition
of I.
- 69.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fa6da54-4e28-4e8d-979d-89b29b7ed7e0%2Fb5198d4d-a8ce-4927-84f9-0b60770ebe59%2F1dptwcu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3 Eigenvalues and Eigenvectors
Represent the matrix A as A = V-¹AV, where A = diag(A... An) is the diagonal matrix
with the eigenvalues X EC of A on its diagonal, and V = [...] is the matrix whose
columns are the eigenvectors, such that , E C is an eigenvector associated with the eigen-
value A. Whenever possible, normalize the eigenvector so that the first coordinate of the
vector is 1 (if the first coordinate is 0, normalize it so that the second coordinate is 1).
(a) A=
-5 3
-18 10
(b) A =
1.
Calculate A² and A²-3A +27. Calculate trace(A) and det(A).
(c) A =
(Can you conclude what is A-¹ without any additional calculation?)
3 -0.5
(d) A=
(e) A =
10
-24 10
10] Calculate A² and A²-41. Calculate trace(A) and det(A).
.
(f) A =
-41-
... mind the complex values. Calculate A²-2A +21. Calculate trace(4)
and det(A).
(8) 4 = 41¹]. (This matrix represents contra-clockwise rotation by 45 degrees.)
If you like extra practice, you can find the decomposition using the following trick:
First calculate A², this is the matrix representing the contra-clockwise rotation by 90
degrees. Represent A² as V-¹AV, and then A = V-¹AV, where A solves the equation
A² = A (simply take square root of each entry on the diagonal, however, keep in mind
that both √ and -√ solve the equation X² = Ã; remember that Vi= (1 + i) and
i=(1-i)). You should then verify that indeed A = V-¹AV, because in fact there
are 4 matrices A such that A² = Ã. Note that this trick is not universal, for example
AS = I, but you won't be able to find the decomposition of A using the decomposition
of I.
- 69.
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