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wlad
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Is the following set of square real square matrices dense: Those that can be diagonalised by a square root of the identity

It's well known that the following set of square real square matrices is dense: those matrices $M$ for which there exists an invertible matrix $P$ such that $P M P^{-1}$ is diagonal. My question is can this statement be strengthened so that $P = P^{-1}$?

The motivation for this question comes from the spectral theorem, whose statement is: Given a matrix $M$ such that $M = M^T$, there exists a matrix $P$ such that $P M P^{-1}$ is diagonal and $P P^T = I$. I'm wondering what happens if you drop the transpose operation. The resulting statement is clearly false, but the question remains "how true" it is.

Note that the set of matrices $P$ such that $P=P^{-1}$ is the same as the set of matrices that are diagonalisable and whose eigenvalues are $\pm 1$.

Is the following set of square real matrices dense: Those that can be diagonalised by a square root of the identity

It's well known that the following set of square real matrices is dense: those matrices $M$ for which there exists an invertible matrix $P$ such that $P M P^{-1}$ is diagonal. My question is can this statement be strengthened so that $P = P^{-1}$?

The motivation for this question comes from the spectral theorem, whose statement is: Given a matrix $M$ such that $M = M^T$, there exists a matrix $P$ such that $P M P^{-1}$ is diagonal and $P P^T = I$. I'm wondering what happens if you drop the transpose operation. The resulting statement is clearly false, but the question remains "how true" it is.

Note that the set of matrices $P$ such that $P=P^{-1}$ is the same as the set of matrices that are diagonalisable and whose eigenvalues are $\pm 1$.

Is the following set of real square matrices dense: Those that can be diagonalised by a square root of the identity

It's well known that the following set of real square matrices is dense: those matrices $M$ for which there exists an invertible matrix $P$ such that $P M P^{-1}$ is diagonal. My question is can this statement be strengthened so that $P = P^{-1}$?

The motivation for this question comes from the spectral theorem, whose statement is: Given a matrix $M$ such that $M = M^T$, there exists a matrix $P$ such that $P M P^{-1}$ is diagonal and $P P^T = I$. I'm wondering what happens if you drop the transpose operation. The resulting statement is clearly false, but the question remains "how true" it is.

Note that the set of matrices $P$ such that $P=P^{-1}$ is the same as the set of matrices that are diagonalisable and whose eigenvalues are $\pm 1$.

edited body; edited title
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wlad
  • 4.9k
  • 2
  • 21
  • 45

Is the following set of square real matrices dense: Those that can be diagonalised by a square root of the identity

It's well known that the following set of real square real matrices is dense: those matrices $M$ for which there exists an invertible matrix $P$ such that $P M P^{-1}$ is diagonal. My question is can this statement be strengthened so that $P = P^{-1}$?

The motivation for this question comes from the spectral theorem, whose statement is: Given a matrix $M$ such that $M = M^T$, there exists a matrix $P$ such that $P M P^{-1}$ is diagonal and $P P^T = I$. I'm wondering what happens if you drop the transpose operation. The resulting statement is clearly false, but the question remains "how true" it is.

Note that the set of matrices $P$ such that $P=P^{-1}$ is the same as the set of matrices that are diagonalisable and whose eigenvalues are $\pm 1$.

Is the following set of square matrices dense: Those that can be diagonalised by a square root of the identity

It's well known that the following set of real square matrices is dense: those matrices $M$ for which there exists an invertible matrix $P$ such that $P M P^{-1}$ is diagonal. My question is can this statement be strengthened so that $P = P^{-1}$?

The motivation for this question comes from the spectral theorem, whose statement is: Given a matrix $M$ such that $M = M^T$, there exists a matrix $P$ such that $P M P^{-1}$ is diagonal and $P P^T = I$. I'm wondering what happens if you drop the transpose operation. The resulting statement is clearly false, but the question remains "how true" it is.

Note that the set of matrices $P$ such that $P=P^{-1}$ is the same as the set of matrices that are diagonalisable and whose eigenvalues are $\pm 1$.

Is the following set of square real matrices dense: Those that can be diagonalised by a square root of the identity

It's well known that the following set of square real matrices is dense: those matrices $M$ for which there exists an invertible matrix $P$ such that $P M P^{-1}$ is diagonal. My question is can this statement be strengthened so that $P = P^{-1}$?

The motivation for this question comes from the spectral theorem, whose statement is: Given a matrix $M$ such that $M = M^T$, there exists a matrix $P$ such that $P M P^{-1}$ is diagonal and $P P^T = I$. I'm wondering what happens if you drop the transpose operation. The resulting statement is clearly false, but the question remains "how true" it is.

Note that the set of matrices $P$ such that $P=P^{-1}$ is the same as the set of matrices that are diagonalisable and whose eigenvalues are $\pm 1$.

Source Link
wlad
  • 4.9k
  • 2
  • 21
  • 45

Is the following set of square matrices dense: Those that can be diagonalised by a square root of the identity

It's well known that the following set of real square matrices is dense: those matrices $M$ for which there exists an invertible matrix $P$ such that $P M P^{-1}$ is diagonal. My question is can this statement be strengthened so that $P = P^{-1}$?

The motivation for this question comes from the spectral theorem, whose statement is: Given a matrix $M$ such that $M = M^T$, there exists a matrix $P$ such that $P M P^{-1}$ is diagonal and $P P^T = I$. I'm wondering what happens if you drop the transpose operation. The resulting statement is clearly false, but the question remains "how true" it is.

Note that the set of matrices $P$ such that $P=P^{-1}$ is the same as the set of matrices that are diagonalisable and whose eigenvalues are $\pm 1$.