Editing Diagonalizable matrix (section)

=== How to diagonalize a matrix ===
Diagonalizing a matrix is the same process as finding its [[eigenvalues and eigenvectors]], in the case that the eigenvectors form a basis. For example, consider the matrix

:<math>A=\left[\begin{array}{rrr}
0 & 1 & \!\!\!-2\\
0 & 1 & 0\\
1 & \!\!\!-1 & 3
\end{array}\right].</math>

The roots of the [[characteristic polynomial]] <math>p(\lambda)=\det(\lambda I-A)</math> are the eigenvalues {{nowrap|<math>\lambda_1 = 1,\lambda_2 = 1,\lambda_3 = 2</math>.}} Solving the linear system <math>\left(1I-A\right) \mathbf{v} = \mathbf{0}</math> gives the eigenvectors <math>\mathbf{v}_1 = (1,1,0)</math> and {{nowrap|<math>\mathbf{v}_2 = (0,2,1)</math>,}} while <math>\left(2I-A\right)\mathbf{v} = \mathbf{0}</math> gives {{nowrap|<math>\mathbf{v}_3 = (1,0,-1)</math>;}} that is, <math>A \mathbf{v}_i = \lambda_i \mathbf{v}_i</math> for {{nowrap|<math>i = 1,2,3</math>.}} These vectors form a basis of {{nowrap|<math>V = \mathbb{R}^3</math>,}} so we can assemble them as the column vectors of a [[Change of basis|change-of-basis]] matrix <math>P</math> to get:
<math display="block">P^{-1}AP =
\left[\begin{array}{rrr}
1 & 0 & 1\\
1 & 2 & 0\\
0 & 1 & \!\!\!\!-1
\end{array}\right]^{-1}

\left[\begin{array}{rrr}
0 & 1 & \!\!\!-2\\
0 & 1 & 0\\
1 & \!\!\!-1 & 3
\end{array}\right]

\left[\begin{array}{rrr}
1 & \,0 & 1\\
1 & 2 & 0\\
0 & 1 & \!\!\!\!-1
\end{array}\right]
=
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix} = D .</math>
We may see this equation in terms of transformations: <math>P</math> takes the standard basis to the eigenbasis, {{nowrap|<math>P \mathbf{e}_i = \mathbf{v}_i</math>,}} so we have:
<math display="block">P^{-1} AP \mathbf{e}_i =
P^{-1} A \mathbf{v}_i =
P^{-1} (\lambda_i\mathbf{v}_i) =
\lambda_i\mathbf{e}_i,</math>
so that <math>P^{-1} AP</math> has the standard basis as its eigenvectors, which is the defining property of {{nowrap|<math>D</math>.}} 

Note that there is no preferred order of the eigenvectors in {{nowrap|<math>P</math>;}} changing the order of the [[eigenvectors]] in <math>P</math> just changes the order of the [[eigenvalues]] in the diagonalized form of {{nowrap|<math>A</math>.}}<ref>{{cite book| last1=Anton |first1=H. |last2= Rorres|first2= C. |title=Elementary Linear Algebra (Applications Version) | url=https://archive.org/details/studentsolutions00grob | url-access=registration |publisher=John Wiley & Sons|edition=8th|date=22 Feb 2000| ISBN= 978-0-471-17052-5}}</ref>