Show that there exists a unique non-zero polynomial PB E R[X] with leading coefficient 1 such that I (PB). This polynomial PB is called the minimal polynomial of the matrix B.
Show that there exists a unique non-zero polynomial PB E R[X] with leading coefficient 1 such that I (PB). This polynomial PB is called the minimal polynomial of the matrix B.
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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send solution for part b only handwritten solution accepted
![Problem 3. Consider the ring M3(R) of 3 x 3 real matrices. As usual, we denote by 03 the
zero-matrix and by I; the identity matrix.
For a matrix A e M3(R) and a polynomial
P(X) = anX" + an-1X"-1+...+ a1X + ao E R[X]
%3D
with ao, ..., a, E R, we define the following matrix:
P(A) := an A" + an-1A"-1 + ... + a1A + aola E M3(R),
and we define the following set:
IA := {P € R[X] such that P(A) 03}.
This set is an ideal of R[X]. (You do not have to prove this fact. A similar result
was seen in the course.)
In this problem, we study the following matrix:
10 1 0
B:= 0 0 1 E M3(R).
0 0 0
(a) Show that the polynomial X E R[X] is an element of the ideal Ig.
(b) Show that there exists a unique non-zero polynomial PB E R[X] with leading
coefficient 1 such that I (PB). This polynomial PB is called the minimal
polynomial of the matrix B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7068779e-a624-45ba-8c15-d1ec8879533a%2F52a2bab4-9cdd-4a62-af6c-aa5a84a3a861%2Fw0pud59_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 3. Consider the ring M3(R) of 3 x 3 real matrices. As usual, we denote by 03 the
zero-matrix and by I; the identity matrix.
For a matrix A e M3(R) and a polynomial
P(X) = anX" + an-1X"-1+...+ a1X + ao E R[X]
%3D
with ao, ..., a, E R, we define the following matrix:
P(A) := an A" + an-1A"-1 + ... + a1A + aola E M3(R),
and we define the following set:
IA := {P € R[X] such that P(A) 03}.
This set is an ideal of R[X]. (You do not have to prove this fact. A similar result
was seen in the course.)
In this problem, we study the following matrix:
10 1 0
B:= 0 0 1 E M3(R).
0 0 0
(a) Show that the polynomial X E R[X] is an element of the ideal Ig.
(b) Show that there exists a unique non-zero polynomial PB E R[X] with leading
coefficient 1 such that I (PB). This polynomial PB is called the minimal
polynomial of the matrix B.
![(c) Show that, in R[X], the only divisors of X3 with leading coefficient 1 are 1,
X, X², and X3.
(d) Deduce from the previous parts that the minimal polynomial of B is X. Is
that polynomial irreducible in R[X]?
(e) The previous parts show that the minimal polynomial of a matrix is not
always irreducible. By contrast, we saw in the course that the minimal
polynomial of an algebraic number is always irreducible. What difference
between the two situations (ring M3(R) of matrices and ring C of complex
numbers) explains this different behaviour?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7068779e-a624-45ba-8c15-d1ec8879533a%2F52a2bab4-9cdd-4a62-af6c-aa5a84a3a861%2Fzyq9nrp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(c) Show that, in R[X], the only divisors of X3 with leading coefficient 1 are 1,
X, X², and X3.
(d) Deduce from the previous parts that the minimal polynomial of B is X. Is
that polynomial irreducible in R[X]?
(e) The previous parts show that the minimal polynomial of a matrix is not
always irreducible. By contrast, we saw in the course that the minimal
polynomial of an algebraic number is always irreducible. What difference
between the two situations (ring M3(R) of matrices and ring C of complex
numbers) explains this different behaviour?
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