Find the eigenvalues and eigenvectors of a matrix

Problem, given the matrix \[ \left ( {\begin {array} [c]{ccc}1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end {array}} \right ) \] Find its eigenvalues and eigenvectors.

Remove["Global`*"] 
(a = {{1,2,3}, {4,5,6}, {7,8,9}}) 
   // MatrixForm

\[ \left ( {\begin {array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end {array}} \right ) \]

Eigenvalues[a] 
N[%]

\[ \left \{\frac {3}{2} \left (5+\sqrt {33}\right ),\frac {3}{2} \left (5-\sqrt {33}\right ),0\right \} \] \[ \{16.1168,-1.11684,0.\} \]

Eigenvectors[a] 
N[%]

\[ \left ( {\begin {array}{ccc} -\frac {-15-\sqrt {33}}{33+7 \sqrt {33}} & \frac {4 \left (6+\sqrt {33}\right )}{33+7 \sqrt {33}} & 1 \\ -\frac {15-\sqrt {33}}{-33+7 \sqrt {33}} & \frac {4 \left (-6+\sqrt {33}\right )}{-33+7 \sqrt {33}} & 1 \\ 1 & -2 & 1 \\ \end {array}} \right ) \] \[ \left ( {\begin {array}{ccc} 0.283349 & 0.641675 & 1. \\ -1.28335 & -0.141675 & 1. \\ 1. & -2. & 1. \\ \end {array}} \right ) \]

Matlab generated eigenvectors are such that the sum of the squares of the eigenvector elements add to one.

clear all; close all; 
 
A=[1 2 3;4 5 6;7 8 9]; 
 
[v,e]=eig(A)

v = 
   -0.2320   -0.7858    0.4082 
   -0.5253   -0.0868   -0.8165 
   -0.8187    0.6123    0.4082 
 
e = 
   16.1168         0         0 
         0   -1.1168         0 
         0         0   -0.0000

First vector shows eigenvalues, and matrix on the right shows the eigenvectors in same order. \[ \left [ \left [\begin {array}{c} 16.1168439700 \\ - 1.1168439700 \\ 0.0 \end {array}\right ] , \left [\begin {array}{ccc} 0.2833494518 & - 1.2833494520 & 1.0 \\ 0.6416747260 & - 0.1416747258 & - 2.0 \\ 1.0 & 1.0 & 1.0 \end {array}\right ] \right ] \]

1.22 Find the eigenvalues and eigenvectors of a matrix