Diagonal Matrix

Description:

• Diagonal matrices are square matrices A with $A_{ij}=0$ when $i\ne j$
• A diagonal $n\times n$ matrix $A$ can be denoted as $A=(a)$, with $a\in {\mathbb{R}}^n$ the vector containing the elements on the diagonal. We can also write
  • $A=\left(\begin{array}{ccc}{a}_1& & \\ & \ddots & \\ & & a_r\end{array}\right)$
  • where by convention the zeros outside the diagonal are not written.

Diagonalizable matrix:

• theorem
  • Let ${\lambda }_i,i=1,...,k\le n$, be the distinct Eigenvalue of $A\in {\mathbb{R}}^{n,n}$
  • Let ${\mu }_i,i=1,...,k$, denote the corresponding algebraic multiplicities (number of times the Eigenvalue is repeated)
  • Let ${\varphi }_i=\mathcal{N}({\lambda }_i{I}_n-A)=\mathcal{N}(A-{\lambda }_i{I}_n)$
    • Meaning find the (x,y,z)s that satisfy $(A-{\lambda }_i{I}_n).(xyz{)}^{⊺}=(000{)}^{⊺}$
    • each solution is a basis of $\varphi$ which is $u$ in the solution
    • Note that ${\mu }_i$ also denotes how many different set of solutions exist
  • Let $U^{(i)}=[u_1^{(i)}\cdot \cdot \cdot u_{\nu i}^{(i)}]$ be a matrix containing by columns a basis of ${\varphi }_i$ , being ${\nu }_i\doteq \dim {\varphi }_i$
  • It holds that ${\nu }_i\le {\mu }_i$ and, if ${\nu }_i={\mu }_i,i=1,...,k$, then $U=[U^{(1)}\cdot \cdot \cdot U^{(k)}]$ is invertible, and $A=U\Lambda U^{-1}$
  • where $\Lambda =\left[\begin{array}{cccc}{\lambda }_1{I}_{\mu 1}& 0& ...& 0\\ 0& {\lambda }_2{I}_{\mu 2}& ...& 0\\ ...& ...& ...& ...\\ 0& ...& 0& {\lambda }_k{I}_{\mu k}\end{array}\right]$