Tridiagonalize

Matrix.Tridiagonalize(M) returns the matrix in triple-diagonal form. The first element returned is a try-diagonal response function representing the matrix. The second element returned is a matrix that represents the unitary matrix that transforms the Hermitian matrix M to try-diagonal form. Possible options include

• BlockSize - the size of the blocks in the tri-diagonal matrix
• NTri - the maximal number of tridiagonal blocks
• SingularValue - the minimum singular value that is considered to be different from 0 of Matrix M.

###

===== Example =====

==== Input ====

Example.Quanty

M = Matrix.New({{1,1,1,1,1,1},
                {1,1,2,3,4,5},
                {1,2,3,4,8,6},
                {1,3,4,5,6,7},
                {1,4,8,6,7,8},
                {1,5,6,7,8,1}})
G1, U1 = Matrix.Tridiagonalize(M,{{"BlockSize",1}})
G2, U2 = Matrix.Tridiagonalize(M,{{"BlockSize",2}})
 
print("The matrix M in tri-diagonal form is")
print(ResponseFunction.ToMatrix(G1))
print("The unitary matrix U1 that transforms M to tridiagonal form is given as")
print(U2)
print("And if we look at U^**M*U^T we get")
print(Chop(U1*M*Matrix.Transpose(U1)))
 
print("The matrix M in Block tri-diagonal form is")
print(ResponseFunction.ToMatrix(G2))
print("The unitary matrix U2 that transforms M to tridiagonal form is given as")
print(U2)
print("And if we look at U^**M*U^T we get")
print(Chop(U2*M*Matrix.Transpose(U2)))

==== Result ====

The matrix M in tri-diagonal form is
{ {  1      , -2.2361 ,  0      ,  0      ,  0      ,  0      } ,
  { -2.2361 ,  24.6   ,  5.8515 ,  0      ,  0      ,  0      } ,
  {  0      ,  5.8515 , -0.7402 , -2.334  ,  0      ,  0      } ,
  {  0      ,  0      , -2.334  , -2.4962 ,  2.2758 ,  0      } ,
  {  0      ,  0      ,  0      ,  2.2758 , -3.2905 ,  2.162  } ,
  {  0      ,  0      ,  0      ,  0      ,  2.162  , -1.0732 } }
 
The unitary matrix U1 that transforms M to tridiagonal form is given as
{ {  1      ,  0      ,  0      ,  0      ,  0      ,  0      } ,
  {  0      ,  1      ,  0      ,  0      ,  0      ,  0      } ,
  {  0      ,  0      ,  0.8328 ,  0.4502 ,  0.0676 , -0.315  } ,
  {  0      ,  0      ,  0.0805 ,  0.312  ,  0.5435 ,  0.7751 } ,
  {  3e-17  ,  0      ,  0.4415 , -0.8303 ,  0.3361 ,  0.0527 } ,
  { -8e-17  ,  0      , -0.3242 ,  0.1032 ,  0.7662 , -0.5452 } }
 
And if we look at U^**M*U^T we get
{ {  1      , -2.2361 ,  0      ,  0      ,  0      ,  0      } ,
  { -2.2361 ,  24.6   ,  5.8515 ,  0      , -3e-15  ,  0      } ,
  {  0      ,  5.8515 , -0.7402 , -2.334  ,  0      ,  0      } ,
  {  0      ,  0      , -2.334  , -2.4962 ,  2.2758 ,  0      } ,
  {  0      , -3e-15  ,  0      ,  2.2758 , -3.2905 ,  2.162  } ,
  {  0      ,  0      ,  0      ,  0      ,  2.162  , -1.0732 } }
 
The matrix M in Block tri-diagonal form is
{ {  1      ,  1      ,  1.0355 ,  1.7111 ,  0      ,  0      } ,
  {  1      ,  1      ,  1.7111 ,  7.1465 ,  0      ,  0      } ,
  {  1.0355 ,  1.7111 ,  2.0149 ,  11.884 ,  1.062  ,  0.4744 } ,
  {  1.7111 ,  7.1465 ,  11.884 ,  16.985 ,  0.4744 ,  3.0368 } ,
  {  0      ,  0      ,  1.062  ,  0.4744 ,  0.8681 ,  0.9176 } ,
  {  0      ,  0      ,  0.4744 ,  3.0368 ,  0.9176 , -3.8681 } }
 
The unitary matrix U2 that transforms M to tridiagonal form is given as
{ {  1      ,  0      ,  0      ,  0      ,  0      ,  0      } ,
  {  0      ,  1      ,  0      ,  0      ,  0      ,  0      } ,
  {  0      ,  0      ,  0.8328 ,  0.4502 ,  0.0676 , -0.315  } ,
  {  0      ,  0      ,  0.0805 ,  0.312  ,  0.5435 ,  0.7751 } ,
  {  0      ,  0      ,  0.4415 , -0.8303 ,  0.3361 ,  0.0527 } ,
  {  0      ,  0      , -0.3242 ,  0.1032 ,  0.7662 , -0.5452 } }
 
And if we look at U^**M*U^T we get
{ {  1      ,  1      ,  1.0355 ,  1.7111 ,  0      ,  0      } ,
  {  1      ,  1      ,  1.7111 ,  7.1465 ,  0      ,  0      } ,
  {  1.0355 ,  1.7111 ,  2.0149 ,  11.884 ,  1.062  ,  0.4744 } ,
  {  1.7111 ,  7.1465 ,  11.884 ,  16.985 ,  0.4744 ,  3.0368 } ,
  {  0      ,  0      ,  1.062  ,  0.4744 ,  0.8681 ,  0.9176 } ,
  {  0      ,  0      ,  0.4744 ,  3.0368 ,  0.9176 , -3.8681 } }

===== Table of contents =====