Diagnolize Calculator 2×2 Matrix

2×2 Matrix Diagonalization Calculator

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What is Matrix Diagonalization?

Matrix diagonalization is the process of converting a square matrix into a diagonal matrix. Diagonal matrices have non-zero entries only on the main diagonal, with all other entries being zero.

Steps to Diagonalize a Matrix:

  1. Find the eigenvalues of the matrix.
  2. Find the corresponding eigenvectors for each eigenvalue.
  3. Construct the matrix of eigenvectors and its inverse.
  4. Form the diagonal matrix using the eigenvalues.
  5. Verify the diagonalization by computing \( P^{-1}AP \), where \( P \) is the matrix of eigenvectors.


What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are concepts in linear algebra. Given a square matrix, an eigenvalue is a scalar that represents how the matrix stretches or shrinks a corresponding eigenvector.

When can a matrix be diagonalized?

A square matrix can be diagonalized if it has \( n \) linearly independent eigenvectors, where \( n \) is the size of the matrix.

What if a matrix has complex eigenvalues?

If a matrix has complex eigenvalues, it cannot be diagonalized using real numbers. However, it can still be diagonalized using complex numbers.

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