# Is Every Matrix Similar to Its Transpose?

• arthurhenry
In summary, it can be shown that two matrices are similar if and only if the Jordan normal form has the same "invariant factors." Additionally, it can be shown that a matrix and its transpose have the same invariant factors. Therefore, it is true that every matrix is similar to its transpose. This is discussed in the book "Matrix Analysis" by Horn and Johnson.
arthurhenry
Is it true that every matrix is similar to its transpose? A claim in Wikipedia...

(field is alg. closed)

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Not every matrix. Only specific types.
I suspect there is a context mentioned?

Whether the base field is algebraically closed doesn't matter here, since two matrices are similar over a small field if and only if they are similar over a bigger field. So the question can always be reduced to an algebraically closed field.

That said, it can be shown that two matrices are similar if and only if the Jordan normal form has the same "invariant factors" and it can be shown that a matrix and it's transpose have the same invariant factors. So a matrix is similar to it's transpose.

Check out the book "matrix analysis" by Horn and Johnson.

I thank you Micromass, that source was very helpful.

## 1. Can a statement be both true and false at the same time?

Yes, in the context of mathematical logic, a statement can be both true and false at the same time. This is known as a contradiction and is typically denoted as "A and not A" or "A ~Trans(A)". However, in everyday language and reasoning, a statement cannot be both true and false simultaneously.

## 2. How is "A ~Trans(A)" different from "A implies ~A"?

"A ~Trans(A)" and "A implies ~A" are two different statements with different meanings. "A ~Trans(A)" means that A is not equivalent to its negation, while "A implies ~A" means that if A is true, then its negation must also be true. The former is a statement about the logical relationship between A and its negation, while the latter is a conditional statement.

## 3. Can a statement be true in one context but false in another?

Yes, the truth value of a statement can vary depending on the context in which it is evaluated. For example, the statement "It is raining" can be true in one location but false in another location. Similarly, the statement "A ~Trans(A)" can be true in one logical system but false in another.

## 4. How is "A ~Trans(A)" related to the law of non-contradiction?

"A ~Trans(A)" is essentially the negation of the law of non-contradiction, which states that a statement cannot be both true and false at the same time. In other words, "A ~Trans(A)" is equivalent to "not(A and not A)", which is the logical form of the law of non-contradiction.

## 5. Can "A ~Trans(A)" be proven or disproven?

"A ~Trans(A)" is a statement that falls under the realm of mathematical logic and cannot be proven or disproven in the same way that empirical claims can be. Instead, it can only be evaluated based on the logical rules and axioms of the system in which it is being considered. In some logical systems, "A ~Trans(A)" may be considered a valid statement, while in others it may be considered contradictory.

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