# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 56: Unitary Linear Operators

In linear algebra, a unitary operator $T$ over a inner product space $V$ is one which satisfies

$TT^* = T^*T = I$.

Thus it is a special kind of normal operator. The following conditions are equivalent to $T$ being unitary:

• $T$ preserves the inner product. That is, $\langle T(x), T(y) \rangle = \langle x, y \rangle$.
• $T$ is distance-preserving. That is, $||T(x)|| = ||x||$.
• $T^*$ is unitary.
• $T$ is invertible and $T^{-1} = T^*$.
• $T$ is a normal operator with eigenvalues on the complex unit circle.

and the following are additionally true of a unitary operator:

• $T$ is normal.
• The eigenspaces of $T$ are orthogonal.
• Every eigenvalue of $T$ has an absolute value of $1$.
• $U = P^*DP$ for some unitary transformations $D, P$, where $D$ is diagonal.

(When $T$ is over $\mathbb{R}$ it is sometimes referred to as ‘orthogonal’ rather than ‘unitary’.)