) What is the algebraic and geometric multiplicities of its eigenvalues. k) Show that the matrix is diagonalizable and find an invertible matrix P and a diagonal matrix D such that P-¹AP = D :) Find a formula for A".
) What is the algebraic and geometric multiplicities of its eigenvalues. k) Show that the matrix is diagonalizable and find an invertible matrix P and a diagonal matrix D such that P-¹AP = D :) Find a formula for A".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Solve remaining parts solution of ist 3 is given
![Given Matrix A =
(a) The characteristic polynomial of matrix A
b(x)= |A-AII = 0
-2
- 112^
1-X
=
(b) The roat of the characteristic
of the matrix
Thus,
= |A²-2A +5
(1-A) ² + 4
1+1² - 2x + 4
Thus,
~[2
~[
2i
12/²
Thus
Step 3: Find the solution
c) Far ₁ = 1-2i
[A- (1-2i)I] x = 0
1² - 2x + 5 = 0
λ =
A
1 ± 2i
the eigenvalues of the matrix A are
A₁ = 1-2i
and 1₂ = 1 + 2i
=) X =
X =
+2√2)² - 4 (1)(5)
2(1)
2 ± 4i
2
Let x₂ = t
the eigen vectors is given by
30-8
x₁ + ix₂ = 0
= -1x₂
Solution
21
JA
J[G]
-i
equation
[:]
then x₁ = -it
Ft+] +[1]
ligenvalue
A₁ = 1-2i.
Far 1₂ = 1+2i the eigenvectors is given by
[A- (1+2i)I]X = 0
-2i -2
-21
[20-8
~ 30-01
2
[JN D
JA
ix2
:: x₁ - x₂ = 0 2)
Let x₂ = t then x₁ = it
t
*- [++) = + [1]
is the basis for the eigenspace corresponding to the
eigenvalues x ₂ = 1 + 2i.
is given by
is called eigenvalues.
is the basis for the ligenspace carrosponding to the
Thus, we get
(a) The characteristic polynomial of matrix A is 1² - 2x+5.
(b) The eigenvalues of matrix A are
1₁ =1-21 and 2 = 1+2i.
(c) The basis for the eigenspaces of the matrix A are
[]] •
and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed5788d2-a0b4-4eb1-998c-a4248e13736e%2F46f74a0b-3f2c-4283-9147-24babdbb7b3b%2Fxifuejh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Given Matrix A =
(a) The characteristic polynomial of matrix A
b(x)= |A-AII = 0
-2
- 112^
1-X
=
(b) The roat of the characteristic
of the matrix
Thus,
= |A²-2A +5
(1-A) ² + 4
1+1² - 2x + 4
Thus,
~[2
~[
2i
12/²
Thus
Step 3: Find the solution
c) Far ₁ = 1-2i
[A- (1-2i)I] x = 0
1² - 2x + 5 = 0
λ =
A
1 ± 2i
the eigenvalues of the matrix A are
A₁ = 1-2i
and 1₂ = 1 + 2i
=) X =
X =
+2√2)² - 4 (1)(5)
2(1)
2 ± 4i
2
Let x₂ = t
the eigen vectors is given by
30-8
x₁ + ix₂ = 0
= -1x₂
Solution
21
JA
J[G]
-i
equation
[:]
then x₁ = -it
Ft+] +[1]
ligenvalue
A₁ = 1-2i.
Far 1₂ = 1+2i the eigenvectors is given by
[A- (1+2i)I]X = 0
-2i -2
-21
[20-8
~ 30-01
2
[JN D
JA
ix2
:: x₁ - x₂ = 0 2)
Let x₂ = t then x₁ = it
t
*- [++) = + [1]
is the basis for the eigenspace corresponding to the
eigenvalues x ₂ = 1 + 2i.
is given by
is called eigenvalues.
is the basis for the ligenspace carrosponding to the
Thus, we get
(a) The characteristic polynomial of matrix A is 1² - 2x+5.
(b) The eigenvalues of matrix A are
1₁ =1-21 and 2 = 1+2i.
(c) The basis for the eigenspaces of the matrix A are
[]] •
and
Transcribed Image Text:(3)
(₁
<s) Consider matrix A =
rk) Find the characteristic polynomial of matrix A.
) Find eigenvalues of the matrix A.
:) Find a basis for the eigenspaces of matrix A.
) What is the algebraic and geometric multiplicities of its eigenvalues.
14
k) Show that the matrix is diagonalizable and find an invertible matrix P and
a diagonal matrix D such that P-¹AP = D
.) Find a formula for A".
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