/Linear_Algebra

Primary LanguagePython

Linear_Algebra

This repository consists of the implementation of 2 projects of Linear Algebra:

  1. Design a program to find the projection matrix P onto the space spanned by 2 vectors.
  2. Design a program to check which of the matrices cannot be diagonalised.

ALGORITHM:

To find the projection matrix P onto the space spanned by 2 vectors:

  1. Give 2 input vectors whose projection matrix has to be found.
  2. Find the transpose of those matrices.
  3. Then find the dot product of obtained transpose with the matrices.
  4. Similarly repeat these steps to obtain projection matrix P given by P = A * (A * AT) AT

To check which of the matrices cannot be diagonalised:

  1. Let A be the n×n matrix that you want to diagonalize.
  2. Find the characteristic polynomial p(t) of A.
  3. Find eigenvalues λ of the matrix A and their algebraic multiplicities from the characteristic polynomial p(t).
  4. For each eigenvalue λ of A, find a basis of the eigenspace Eλ. If there is an eigenvalue λ such that the geometric multiplicity of λ, dim(Eλ), is less than the algebraic multiplicity of λ, then the matrix A is not diagonalizable. If not, A is diagonalizable, and proceed to the next step.
  5. If we combine all basis vectors for all eigenspaces, we obtained n linearly independent eigenvectors v1v2...vn.
  6. Define the nonsingular matrix S = [v1v2...vn].
  7. Define the diagonal matrix D, whose (i,i)-entry is the eigenvalue λ such that the i-th column vector vi is in the eigenspace Eλ.
  8. Then the matrix A is diagonalized as S−1AS = D.