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R.P.
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An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$$A^{\top}$: elementary proof?

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edit approved Oct 9, 2017 at 12:07
Rodrigo de Azevedo
edit approved Oct 9, 2017 at 12:07
Rodrigo de Azevedo
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Matrix Transpose Similarility An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$

A famous problemresult in linear algebra is that "A $n\times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$the following."

An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$.

I know one proof using the Smith Normal formForm (SNF). However, I want to knowfind an elementary proof avoiding any concepts related to the SNF.

  My question is: "Is is there an elementary way to prove this?"

RequirementsThe requirements are:

Do NOT use the structure theorem over PID.

Do NOT use the Smith Normal form (nor Jordan canonical form).

Do NOT use the concept of invariant factors.

Provide an explicit invertible matrix P such that $$A=PA^T P^{-1}.$$

  • Do NOT use the structure theorem over PID.

  • Do NOT use the Smith Normal form (nor Jordan canonical form).

  • Do NOT use the concept of invariant factors.

  • Provide an explicit invertible matrix $P$ such that $A=PA^T P^{-1}$.

Matrix Transpose Similarility

A famous problem in linear algebra is that "A $n\times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$."

I know one proof using Smith Normal form. However, I want to know an elementary proof avoiding any concepts related to SNF.

  My question is: "Is there an elementary way to prove this?"

Requirements are:

Do NOT use the structure theorem over PID.

Do NOT use the Smith Normal form (nor Jordan canonical form).

Do NOT use the concept of invariant factors.

Provide an explicit invertible matrix P such that $$A=PA^T P^{-1}.$$

An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$

A famous result in linear algebra is the following.

An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$.

I know one proof using the Smith Normal Form (SNF). However, I want to find an elementary proof avoiding any concepts related to the SNF. My question is: is there an elementary way to prove this?

The requirements are:

  • Do NOT use the structure theorem over PID.

  • Do NOT use the Smith Normal form (nor Jordan canonical form).

  • Do NOT use the concept of invariant factors.

  • Provide an explicit invertible matrix $P$ such that $A=PA^T P^{-1}$.

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Sungjin Kim
Sungjin Kim
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Matrix Transpose Similarility

A famous problem in linear algebra is that "A $n\times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$."

I know one proof using Smith Normal form. However, I want to know an elementary proof avoiding any concepts related to SNF.

My question is: "Is there an elementary way to prove this?"

Requirements are:

Do NOT use the structure theorem over PID.

Do NOT use the Smith Normal form (nor Jordan canonical form).

Do NOT use the concept of invariant factors.

Provide an explicit invertible matrix P such that $$A=PA^T P^{-1}.$$