[Math] Orthogonal, symmetric and positive definite matrices!

linear algebra

I know that the identity matrix is orthogonal, symmetric and positive definite. I would like to know if there are some other matrices that have the three above-mentioned characteristics simultaneously. I guess that the identity matrix is the only such a matrix.

Best Answer

If $A$ is a real $n\times n$ matrix which is orthogonal and symmetric, then $A^2=AA^T=I$. Therefore the eigenvalues of $A$ are all equal to $\pm1$. If also $A$ is positive definite then they must all be equal to $1$.

Finally, $A$ is diagonalizable since it is symmetric, and since all the eigenvalues of $A$ are equal to $1$ it follows that $A$ is the identity matrix.

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