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--- topic:math:linalg:operator [2023/02/13 18:57] – removed - external edit (Unknown date) 127.0.0.1
+++ topic:math:linalg:operator [2023/02/14 17:14] (current) – [Operator (Matrix) properties] samuel
@@ Line 1: / Line 1: @@
+====== Operators ======
+===== Hilbert Space =====
+The **Hilbert Space** is an inner product space, which is complete with respect to the norm defined by the inner product.
+See also [[https://www.quantiki.org/wiki/hilbert-spaces | on Quantiki]]
+===== Operator (Matrix) properties =====
+  * $U^\dagger$ is the adjoint of $U$, and in matrix representation is given by $(U^*)^T$, where $^*$ marks the complex conjugate and $^T$ the transpose.
+  * An operator (matrix) $U$ is said to be normal if $U^\dagger U = U U^\dagger$.
+    * A normal operator is diagonizable (spectral decomposition)
+  * An operator (matrix)  $U$ is said to be unitary if $U^\dagger U = I$.
+  * An operator (matrix)  $U$ is said to be hermitian if $U = U^\dagger$.
+    * A hermitian operator is normal.

Kollektanee - Samuel Gyger