Eigensystem

Matrix.Eigensystem(M) calculates the eigenvalues and eigenvectors of the square matrix M. If M is Hermitian it returns the eigenvalues and a single eigenvector, if M is non-hermitian it returns both the left and right eigenvectors

For an Hermitian matrix

Example.Quanty

A = Matrix.New({{1,2,3},
                {2,3,5},
                {3,5,1}})
val, fun = Eigensystem(A)
print("The eigenvalues are\n",val)
print("The eigenfunctions are\n",fun)
print("The matrix transformed to a diagonal matrix by its eigenfunctions is\n",Chop( Matrix.Conjugate(fun) * A * Matrix.Transpose(fun)) )

The eigenvalues are
{ -3.4339294734789 , -0.23514404390394 , 8.6690735173829 }
The eigenfunctions are
{ {  0.3019 ,  0.5247 , -0.796  } ,
  {  0.8587 , -0.5123 , -0.012  } ,
  {  0.4141 ,  0.6799 ,  0.6052 } }
 
The matrix transformed to a diagonal matrix by its eigenfunctions is
{ { -3.4339 ,  0      ,  0      } ,
  {  0      , -0.2351 ,  0      } ,
  {  0      ,  0      ,  8.6691 } }

For a non-Hermitian matrix

Example.Quanty

A = Matrix.New({{1,1,3},
                {5,3,7},
                {3,5,1}})
val, funL, funR = Eigensystem(A)
print("The eigenvalues are\n",val)
print("The left  eigenfunctions are\n",funL)
print("The right eigenfunctions are\n",funR)
print("The matrix transformed to a diagonal matrix by its eigenfunctions is\n",Chop( Matrix.Conjugate(funL) * A * Matrix.Transpose(funR)) )

WARNING: non hermitian matrix found
Using left and right handed eigensystem
With potential complex eigenvalues
The eigenvalues are
{ -3.8133538424944 , -0.86687641096757 , 9.680230253462 }
The left  eigenfunctions are
{ {  0.139          , -0.6419         ,  0.847          } ,
  { -0.974          ,  0.4605         , -0.1613         } ,
  { -0.5567         , -0.5736         , -0.655          } }
 
The right eigenfunctions are
{ { -0.3991         , -0.5551         ,  0.8254         } ,
  { -0.9178         ,  0.3854         ,  0.4427         } ,
  { -0.2901         , -0.8128         , -0.5684         } }
 
The matrix transformed to a diagonal matrix by its eigenfunctions is
{ { -3.8134 ,  0      ,  0      } ,
  {  0      , -0.8669 ,  0      } ,
  {  0      ,  0      ,  9.6802 } }

For an Hermitian matrix with small non-Hermitian part

Example.Quanty

A = Matrix.New({{1,1,3},
                {1-1E-7,3,5},
                {3,5,1}})
val, funL, funR = Eigensystem(A)
print("The eigenvalues are\n",val)
print("The left  eigenfunctions are\n",funL)
print("The right eigenfunctions are\n",funR)
print("The matrix transformed to a diagonal matrix by its eigenfunctions is\n",Chop( Matrix.Conjugate(funL) * A * Matrix.Transpose(funR)) )
 
print("The left and right hand vectors are now only marginally different\n",Chop( Matrix.Conjugate(funL) * A * Matrix.Transpose(funL)) )

WARNING: non hermitian matrix found
Using left and right handed eigensystem
With potential complex eigenvalues
The eigenvalues are
{ -3.7873476689872 , 0.64886022722351 , 8.1384874417637 }
The left  eigenfunctions are
{ { -0.3763         , -0.5129         ,  0.7715         } ,
  { -0.853          ,  0.5169         , -0.0725         } ,
  { -0.3616         , -0.6854         , -0.632          } }
 
The right eigenfunctions are
{ { -0.3763         , -0.5129         ,  0.7715         } ,
  { -0.853          ,  0.5169         , -0.0725         } ,
  { -0.3616         , -0.6854         , -0.632          } }
 
The matrix transformed to a diagonal matrix by its eigenfunctions is
{ { -3.7873 ,  0      ,  0      } ,
  {  0      ,  0.6489 ,  0      } ,
  {  0      ,  0      ,  8.1385 } }
 
The left and right hand vectors are now only marginally different
{ { -3.7873 , -5.4e-8 ,  2.3e-9 } ,
  {  9.2e-9 ,  0.6489 , -6.7e-9 } ,
  { -4.9e-9 , -8.4e-8 ,  8.1385 } }