Connecting Unitary, Orthogonal, Normal, and Self-Adjoint I'm trying to connect these 4 properties to see the "big picture," and would love some input or corrections. This is what I have so far:
1) If $T$ is unitary or orthogonal, then $T$ is also normal.
2) Normal operators are also self-adjoint. ($\implies T$ is unitary/orthgonal?) FALSE.
3) Unitary and orthogonal operators have eigenvalues where $|\lambda_i|$ = $1$.
4) Unitary and orthgonal operators are similar to a diagonal matrix (over $\mathbb{C}$ and $\mathbb{R}$, respectively), which means they are diagonalizable.
5) If $T$ is orthogonal, then all of its eigenvalues must be real. FALSE.
6) If $T$ is unitary, not all of its eigenvalues have to be imaginary (can be both complex and real, as long as its absolute values are $1$).
I know not all of these are correct, and I will edit them accordingly. But these are the connections I came up with so far. Any input or corrections would be GREATLY helpful. Thank you!
 A: Here's the big picture:


*

*Normal operators consist of all unitarily diagonalizable operators. The adjoint of a normal operator is normal. Multiplying a normal operator by any number gives you another normal operator. An upper triangular matrix is normal iff it is diagonal.

*The important examples of normal operators are self-adjoint, skew-adjoint and unitary operators.

*A normal operator is self-adjoint iff its eigenvalues are real. A self-adjoint matrix with real entries is symmetric. Symmetric matrices are the only matrices that has real eigenvalues and can be fully diagonalized with orthogonal (real unitary) matrices. A self-adjoint matrix with non negative eigenvalues is positive semidefinite.

*$A$ is self adjoint iff $iA$ is skew adjoint. A skew adjoint matrix with real entries is skew symmetric. A normal matrix is skew adjoint iff it has imaginary eigenvalues.

*A normal matrix is unitary iff its eigenvalues have magnitude $1$. A unitary matrix that happens to have real entries is orthogonal.

*A transformation can be represented as a matrix with real entries with the right choice of basis if and only if its non-real eigenvalues come in conjugate pairs.
