**Theorem 2.1.9.**

If **x**<sup>1</sup> , **x**<sup>2</sup> , ⋯, **x***<sup>k</sup>* are eigenvectors of *A* corresponding to distinct eigenvalues *λ*1, *λ*2, ⋯, *λk*, then {**x**<sup>1</sup> , **x**<sup>2</sup> , ⋯, **x***<sup>k</sup>* } is a linearly independent set.

As consequence of Theorem 2.1.8. and Theorem 2.1.9 ., we obtain the following important result

### **Theorem 2.1.10.**

If an *n* × *n* matrix *A* has *n* distinct eigenvalues, then *A* is diagonalizable.

There are matrices that can have the same eigenvalues and yet can be diagonalizable. Broadest such class of such matrices are normal matrix, which we will introduce the following definition.

**Definition 2.1.5 .** Matrix *A* ∈ ℂ(*n*,*n*) is called normal, if holds *A<sup>H</sup>A* = *AA<sup>H</sup>*.

More general characterization of diagonalizable matrix A is given in the following theorem *.*

#### **Theorem 2.1.11.**

If *A* is a square matrix, then:

