Diagonalization (Noun)
Meaning
Changing a square matrix to diagonal form (with all non-zero elements on the principal diagonal); "the diagonalization of a normal matrix by a unitary transformation".
Classification
Nouns denoting cognitive processes and contents.
Examples
- Diagonalization is a way of converting a matrix to its diagonal form to facilitate further mathematical analysis or calculations.
- By finding the eigenvectors of a square matrix, it is possible to achieve its diagonalization using a similarity transformation.
- The diagonalization of a normal matrix is always guaranteed by using a unitary transformation, which ensures that the resulting matrix is Hermitian and, consequently, diagonalizable.
- The concept of diagonalization plays a crucial role in the analysis of Markov chains as it simplifies the calculation of the steady-state and the transition probabilities.
- In many cases, an efficient diagonalization algorithm can be greatly beneficial in solving systems of linear differential equations or computing eigenvalues of matrices.