4. Determine whether or not the statements below are true. If true, prove it; if false, give a concrete counter-example. 4a. If A is diagonal, then A itself is a Jordan canonical form. 4b. If J is a Jordan canonical form of a linear operator T on V, then J is a Jordan canonical form of the representative matrix [T]B' relative to any basis B' of V. 4c. If A₁ and A2 in Mn(F) have the same characteristic polynomial, then A₁ A2. 4d. If A₁ and A2 have the same Jordan canonical form, then A₁ A2. 4e. Every matrix A € Mn(R) has a a Jordan canonical form J = Mn(R) such that A is similar to J. 4f. Every matrix A € M₂ (F) is similar to its Jordan canonical form J E Mn (F) (if exists). 1
4. Determine whether or not the statements below are true. If true, prove it; if false, give a concrete counter-example. 4a. If A is diagonal, then A itself is a Jordan canonical form. 4b. If J is a Jordan canonical form of a linear operator T on V, then J is a Jordan canonical form of the representative matrix [T]B' relative to any basis B' of V. 4c. If A₁ and A2 in Mn(F) have the same characteristic polynomial, then A₁ A2. 4d. If A₁ and A2 have the same Jordan canonical form, then A₁ A2. 4e. Every matrix A € Mn(R) has a a Jordan canonical form J = Mn(R) such that A is similar to J. 4f. Every matrix A € M₂ (F) is similar to its Jordan canonical form J E Mn (F) (if exists). 1
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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Please help with subparts d,e,f
![4. Determine whether or not the statements below are true. If true, prove
it; if false, give a concrete counter-example.
4a. If A is diagonal, then A itself is a Jordan canonical form.
4b. If J is a Jordan canonical form of a linear operator T on V, then J is
a Jordan canonical form of the representative matrix [T]B' relative to any
basis B' of V.
4c. If A₁ and A2 in Mn(F) have the same characteristic polynomial, then
A₁ A2.
4d. If A₁ and A2 have the same Jordan canonical form, then A₁ A2.
4e. Every matrix A € Mn(R) has a a Jordan canonical form J = Mn(R)
such that A is similar to J.
4f. Every matrix A € M₂ (F) is similar to its Jordan canonical form J E
Mn(F) (if exists).
2
1
4g. If two linear operators T₁ and T₂ on V have the same characteristic
polynomial (x - X)", then they have the same Jordan canonical form.
4h. A linear operator on a finite-dimenional vector space has at most one
Jordan canonical basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fc235f5-c6fd-4e1c-bac8-7d394d2e8123%2Fa996b7c8-3cae-4274-a903-972001153fac%2F4gbauo_processed.png&w=3840&q=75)
Transcribed Image Text:4. Determine whether or not the statements below are true. If true, prove
it; if false, give a concrete counter-example.
4a. If A is diagonal, then A itself is a Jordan canonical form.
4b. If J is a Jordan canonical form of a linear operator T on V, then J is
a Jordan canonical form of the representative matrix [T]B' relative to any
basis B' of V.
4c. If A₁ and A2 in Mn(F) have the same characteristic polynomial, then
A₁ A2.
4d. If A₁ and A2 have the same Jordan canonical form, then A₁ A2.
4e. Every matrix A € Mn(R) has a a Jordan canonical form J = Mn(R)
such that A is similar to J.
4f. Every matrix A € M₂ (F) is similar to its Jordan canonical form J E
Mn(F) (if exists).
2
1
4g. If two linear operators T₁ and T₂ on V have the same characteristic
polynomial (x - X)", then they have the same Jordan canonical form.
4h. A linear operator on a finite-dimenional vector space has at most one
Jordan canonical basis.
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