= Diagonalization = '''Diagonalization''' is an alternative decomposition of square matrices. <> ---- == Description == A '''diagonal matrix''' is a diagonal line of numbers in a square matrix of zeros. Such a matrix has many useful properties. * Its columns are its [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]] * The numbers in the diagonal are the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] * The [[LinearAlgebra/Determinant|determinant]] is the project of the numbers in the diagonal A matrix is '''diagonalizable''' if it can be factored into a diagonal matrix. Only square matrices can be diagonalizable. A square matrix that still cannot be factored as such is '''defective'''. ---- == Process == Given a matrix '''''A''''', notate the matrix of its [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]] as '''''S'''''. A diagonalizable matrix can be factored as '''''A''' = '''SΛS'''^-1^''. '''''Λ''''' will be a diagonal matrix with the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] of '''''A''''' in the diagonal. In other words, '''''A''''' can be rewritten as a eigennormalized (i.e. transformed by '''''S''''') then un-eigennormalized (i.e. transformed by '''''S'''^-1^'') diagonal matrix '''''Λ'''''. ---- == Usage == Diagonalization offers clean solutions to mathematical models. '''''A'''^2^ = '''SΛ'''^2^'''S'''^-1^'', and more generally '''''A'''^K^ = '''SΛ'''^K^'''S'''^-1^''. Similarly, ''e'''^A^''' = '''S'''e'''^Λ^S'''^-1^''. Note that ''e'''^Λ^''''' is a diagonal matrix with ''e'' to the power of the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] of '''''A''''' in the diagonal. ---- CategoryRicottone