If a matrix has positive, real eigenvalues, is it always symmetric?

Question

We know that symmetric matrices are orthogonally diagonalizable and have real eigenvalues. Is the converse true? Does a matrix with real eigenvalues have to be symmetric?

A class of symmetric matrices, the positive definite matrices, have positive real eigenvalues. Is the converse true? Does a matrix with positive real eigenvalues have to be symmetric, positive-definite?

I think the answer to all this is "no", but I just wanted to confirm.

Thanks,

Answer 1

Is the matrix

$$\begin{pmatrix}1&1\\0&1\end{pmatrix}$$

symmetric? It has only one positive eigenvalue of multiplicity two.

Answer 2

No.

The simplest example is any upper triangular matrix whose diagonal entries are real:

$$\begin{pmatrix} 1 & 1-i & 2 \\ 0 & 2 & 3i \\ 0 & 0 & 3 \end{pmatrix}$$

has complex entries in the non-diagonal spots but its eigenvalues are $1,2,3$ and it is certainly not symmetric!

Answer 3

consider on this $$ \begin{bmatrix} 1 & 2 & 2 \\ 0 & 2 & 4 \\ 0 & 0 & 3 \\ \end{bmatrix} $$

Answer 4

The first question seems to be handled quite well by other people but I just want to add one more thing.

Positive-definiteness always starts from symmetry. So the converse can never be true.

That being said, it's only when we have the guarantee that the matrix is symmetric that we can conclude a matrix is positive-definite.