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