I'm reading my linear algebra textbook and there are two sentences that make me confused.
(1) Symmetric matrix $A$ can be factored into $A=Q\lambda Q^{T}$ where $Q$ is orthogonal matrix : Diagonalizable
($Q$ has eigenvectors of $A$ in its columns, and $\lambda$ is diagonal matrix which has eigenvalues of $A$)
(2) Any symmetric matrix has a complete set of orthonormal eigenvectors
whether its eigenvalues are distinct or not.
It's a contradiction, right?
Diagonalizable means the matrix has n distinct eigenvectors (for $n$ by $n$ matrix).
If symmetric matrix can be factored into $A=Q\lambda Q^{T}$, it means that
symmetric matrix has n distinct eigenvalues.
Then why the phrase "whether its eigenvalues are distinct or not" is added in (2)?
After reading eigenvalue and eigenvector part of textbook, I conclude that every symmetric matrix is diagonalizable. Is that true?
Best Answer
Diagonalizable doesn't mean it has distinct eigenvalues. Think about the identity matrix, it is diagonaliable (already diagonal, but same eigenvalues. But the converse is true, every matrix with distinct eigenvalues can be diagonalized.