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Rodrigo de Azevedo
Rodrigo de Azevedo
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I've come across thisthe following question in my lecture notes and am not sure why the answer makes sense. Here's the question:

"Let $f,g \in End(\mathbb{C}^4)$ be neither nilpotent nor invertible with their characteristic and minimal polynomials being equal. Do they have the same Jordan-Form?"

Let $f,g \in End(\mathbb{C}^4)$ be neither nilpotent nor invertible with their characteristic and minimal polynomials being equal. Do they have the same Jordan form?

The answer to this question in my lecture notes is yes which implies that they are similar (correct me if I'm wrong). My theory is that because they have the same characteristic polynomial that means that they have the same eigenvalues with the same algebraic multiplicity. Furthermore, having the same minimal polynomial means that the size of the largest Jordan block is the same for both Jordan-forms. Could someone please expand on this/correct me?

I've come across this question in my lecture notes and am not sure why the answer makes sense. Here's the question:

"Let $f,g \in End(\mathbb{C}^4)$ be neither nilpotent nor invertible with their characteristic and minimal polynomials being equal. Do they have the same Jordan-Form?"

The answer to this question in my lecture notes is yes which implies that they are similar (correct me if I'm wrong). My theory is that because they have the same characteristic polynomial that means that they have the same eigenvalues with the same algebraic multiplicity. Furthermore, having the same minimal polynomial means that the size of the largest Jordan block is the same for both Jordan-forms. Could someone please expand on this/correct me?

I've come across the following question and am not sure why the answer makes sense.

Let $f,g \in End(\mathbb{C}^4)$ be neither nilpotent nor invertible with their characteristic and minimal polynomials being equal. Do they have the same Jordan form?

The answer to this question in my lecture notes is yes which implies that they are similar (correct me if I'm wrong). My theory is that because they have the same characteristic polynomial that means that they have the same eigenvalues with the same algebraic multiplicity. Furthermore, having the same minimal polynomial means that the size of the largest Jordan block is the same for both Jordan-forms. Could someone please expand on this/correct me?

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RakoonBerry
RakoonBerry
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Non-nilpotent and non-invertible matrices that have the same characteristic and minimal polynomials have the same Jordan-form

I've come across this question in my lecture notes and am not sure why the answer makes sense. Here's the question:

"Let $f,g \in End(\mathbb{C}^4)$ be neither nilpotent nor invertible with their characteristic and minimal polynomials being equal. Do they have the same Jordan-Form?"

The answer to this question in my lecture notes is yes which implies that they are similar (correct me if I'm wrong). My theory is that because they have the same characteristic polynomial that means that they have the same eigenvalues with the same algebraic multiplicity. Furthermore, having the same minimal polynomial means that the size of the largest Jordan block is the same for both Jordan-forms. Could someone please expand on this/correct me?