Which of the following statements is NOT true for a square matrix A?
(a)If A is an upper triangular matrix, the eigenvalues of A are the diagonal elements of it.
(b) If A is real symmetric, the eigenvalues of A are always real and positive.
(c) If A is real, the eigenvalues of A and Transpose(A) are always the same.
(d) If all the principal minors of A are positive, all the eigenvalues of A are also positive.
I am confirmed about (a) and (d) but confused between (c) and (b). Please help.
Best Answer
b) Consider the matrix
$$\begin{pmatrix}-1&0\\0&-2\end{pmatrix}$$
(Actually, even $\begin{pmatrix}-1\end{pmatrix}$ is enough.)
c) Diagonalize the matrix as
$$A=PDP^{-1}.$$
Now, $$A^T=P^{-T}D^TP^T=QDQ^{-1}$$ because $D=D^T$.