# 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!

• Skew-symmetric operators are normal, but not self-adjoint. $2\times2$ rotation matrices are orthogonal, but generally don't have real eigenvalues. – Gerry Myerson Dec 13 '16 at 12:22
• 2) and 5) are false – Omnomnomnom Dec 13 '16 at 12:43
• I see why 2) is, but why is 5) false? From my understanding, orthogonal operators deal with real inner product spaces. How can an operator be orthogonal if its eigenvalues are not real? – Max Dec 13 '16 at 12:52
• Oh, would a correction for 5) be that if $T$ is orthogonal and self-adjoint, then its eigenvalues are $|1|$? – Max Dec 13 '16 at 12:58
• If $T$ is orthogonal and self-adjoint, its eigenvalues are $\pm 1$. – Omnomnomnom Dec 13 '16 at 13:08

Here's the big picture:

1. 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.

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

3. 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.

4. $A$ 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.

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

6. 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.