Jacobian: Overview, Questions, Preparation

A matrix is a form of representation of a pair of linear equations. The coefficients of a variable are structured in a box format to form a matrix. For a system of linear equations,
m₁x+ n₁y =k₁ and m₂x+ n₂y =k₂, the matrix representation is:

For a unique solution, the condition is m₁n₂- m₂ n₁ ≠0. This equation is represented in the form of a matrix,

 = M. This matrix is known as the determinant of M.

The determinant of a matrix of partial derivatives is known as the ‘Jacobian’.

The method that is used to determine the solutions of a system of linear equations is known as the Jacobian method. It is an iterative algorithm where the diagonal elements of a matrix are assigned an approximate value. Then, the matrix is solved to reach convergence, and this process of matrix diagonalisation is known as the Jacobi transformation.

Consider a matrix M for the ‘n’ system of linear equations.

Then, Mx = N

m₁₁      m₁₂   .............m_1n

m₂₂     m₂₃    .............m_2n

...      ...          ...         ...

m_n1, m_n2,   ...          m_nn

x= x₁, x₂ ....x_n 

p= p₁, p₂ ....p_n 

_{Decompose the matrix M into the diagonal component ‘D’ and remainder ‘R’ such that M= D+ R.
D=}

m₁₁     0   ...       0 

0         m₂₂ ...     0

...        ....          ....

0          0   ....... m_mn

_R= 

0         m₁₂  ...      m_1n  

m₂₂    0      ...       m_2n 

...        .....               ....

m_n1      m_n2  ....... 0

So, the solution for these equations can be denoted as:
x^(s+1) = D^-1 (b- Rx^s),
Where x^s is the s^th iteration or approximation of s. Also, x^s+1 is the (s+1)^th iteration of x or the next iteration after x^s.