# Matrix diagonalization by eigen decomposition

Posted 5 months ago
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 Hey! Let A be n×n matrix. I am looking for its D eigenvalue- and X eigenvector matrices for which: X×D÷X = A, what's more f(A) = X×f(D)÷X.In this example: A = {{1,2},{3,2}} Let's determine its eigenvectors and -values: {Da, Xa} = Eigensystem[{{1, 2}, {3, 2}}] The output: {{4, -1}, {{2, 3}, {-1, 1}}} where the eigenvalues are correct and the eigenvectors are not! Because: Xa*DiagonalMatrix[Da]/Xa returns {{4, 0}, {0, -1}} which differs to matrix A. I would like to get {{-1,2/3},{1,1}} as eigenvectors. How to solve the problem?(I would like to use it for iteration of bigger Carleman matrices. Thank you!)
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Posted 5 months ago
 One can checkthe documentation for Eigensystem for how to get the diagonalization: "The eigenvalues and eigenvectors satisfy the matrix equation m.Transpose[vectors]==Transpose[vectors].DiagonalMatrix[values]. »"The eigenvectors in this example are quite correct, by the way. Stating otherwise is simply misrepresenting the situation.
 You are right. I just misused the functions. Now it works: Xb = Transpose[Eigenvectors[A]] Db = DiagonalMatrix[Eigenvalues[A]] And Xb.Db.Inverse[Xb] returns A as it should be. Thank you!