: Astronomy: matrix diagonalization

mathematical technique > algebra > vector space > matrix > matrix diagonalization

matrix diagonalization

In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that P⁻¹ A P = D, or equivalently A = P D P⁻¹. (Such P P, D D are not unique.). For a finite-dimensional vector space V, a linear map T : V → V is called diagonalizable if there exists an ordered basis of V consisting of eigenvectors of T. These definitions are equivalent : if T has a matrix representation T = P D P⁻¹ as above, then the column vectors of P form a basis consisting of eigenvectors of T, and the diagonal entries of D are the corresponding eigenvalues of T; with respect to this eigenvector basis, A is represented by D. Diagonalization is the process of finding the above P and D. (Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Diagonalizable_matrix)

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