A normal matrix doesn't mean that all the columns in the matrix are normalized. --- ### **Intro:** When a matrix is normal, it satisfies the following conditions: Let's say that the matrix $A$ is a normal matrix then: #### **Def | A Normal Matrix** > $A^HA = AA^H$, or Equivalently $A = Q\Lambda Q^H$ Where $Q$ is an unitary matrix. If a matrix is similar to a diagonal matrix by a unitary transform, then the matrix is going to be normal. ### **Interesting Fact:** When the matrix is normal, the singular values of the matrix are the absolute eigen values of the original matrix. This is followed from the theorems from SVD, [[SVD Theorems]] --- ### **Normal vs Hermitian** Hermitian Matrix has real eigenvalues and diagonalizable, and they can be written as $Q\Lambda Q^H$, and the diagonals of $\Lambda$ are all positive reals. Normal Matrix can be written as $Q\Lambda Q^H$, where the diagonal of $\Lambda$ can have any kind of numbers it wants (for example, complex numbers). > Hermitian are a special kind of Normal.