L 33 Diagonalization - WordPress
L 33Diagonalization
Diagonalization Diagonalization problem:For a square matrix A, does there exist an invertible matrix Psuch that P-1AP is diagonal? Diagonalizable matrix:A square matrix A is called diagonalizable if there exists aninvertible matrix P such that P-1AP is a diagonal matrix. Notes:(P diagonalizes A)(1) If there exists an invertible matrix P such that B P 1 AP,then two square matrices A and B are called similar.(2) The eigenvalue problem is related closely to thediagonalization problem.
Thm : (Similar matrices have the same eigenvalues)If A and B are similar n n matrices, then they have thesame eigenvalues.Pf:A and B are similar B P 1 AP I B I P 1 AP P 1 IP P 1 AP P 1 ( I A) P P 1 I A P P 1 P I A P 1 P I A I AThus A and B have the same eigenvalues.
Ex 1: (A diagonalizable matrix) check the following matrix is diagonal or not. 1 3 0 A 3 1 0 00 2 Sol: Characteristic equation: 1 30 I A 3 10 ( 4)( 2) 2 000 2The eigenvalue s： 1 4, 2 2, 3 2 1 (1) 4 the eigenvecto r p1 1 0
1 0 (2) 2 the eigenvecto r p2 1 , p3 0 0 1 1 1 0 P [ p1 p2 p3 ] 1 1 0 ， 0 0 1 0 4 0such that P 1 AP 0 2 0 0 0 2 Note: If P [ p2 p1 p3 ] 1 1 0 1 1 0 001 2 0 0 P 1 AP 0 4 0 00 2
Thm : (Condition for diagonalization)An n n matrix A is diagonalizable if and only if it has nlinearly independent eigenvectors.
Ex 4: (A matrix that is not diagonalizable)Show that the following matrix is not diagonaliz able. 1 2 A 01 Sol: Characteristic equation: 1 2 I A ( 1) 2 00 1The eigenvalue ： 1 1 0 2 0 1 1 I A I A eigenvector p1 0 0 0 0 0 A does not have two linearly independent eigenvectors,so A is not diagonalizable.
Steps for diagonalizing an n n square matrix:Step 1: Find n linearly independent eigenvectorsp1 , p2 ,pn for A with corresponding eigenvalues.Step 2: Let P [ p1Step 3:p2pn ]0 1 0 0 02 P 1 AP D n 0 0where, Api i pi , i 1, 2, , n
Thm 5.6: (Sufficient conditions for diagonalization)If an n n matrix A has n distinct eigenvalues, then thecorresponding eigenvectors are linearly independent andA is diagonalizable. Ex 5: (Determining whether a matrix is diagonalizable) 1 2 1 A 0 01 0 0 3 Sol: Because A is a triangular matrix, its eigenvalues are 1 1, 2 0, 3 3These three values are distinct, so A is diagonalizable.
Ex 6: (Diagonalizing a matrix) 1 1 1 A 13 1 3 1 1 Find a matrix P such that P 1 AP is diagonal.Sol: Characteristic equation: 1 I A 1311 3 1 ( 2)( 2)( 3) 0 1 1The eigenvalue s： 1 2, 2 2, 3 3
1 2 1 1 1 1 0 1I A 1 1 1 0 1 3 13 0 0 x1 t 1 x 0 eigenvecto r p 0 1 2 x3 t 1 2 21 0 0 1 1 0 14 3 1 2 I A 1 5 1 0 1 14 3 1 1 0 0 0 x1 14 t 1 x 1 t eigenvecto r p 1 2 2 4 x3 t 4
3 3 2 1 1 1 0 1 3I A 1 0 1 0 1 1 3 14000 x1 t 1 x t eigenvecto r p 1 3 2 x3 t 1 1 1 1 1P [ p1 p2 p3 ] 0 1 1 ， 𝑃 1 1/51/5 1 4 1 2 0 0 s.t. P 1 AP 0 2 0 0 0 3 100 1/51 1/5
Diagonalization Diagonalization problem: For a square matrix A, does there exist an invertible matrix P such that P-1AP is diagonal? Diagonalizable matrix: A square matrix A is called diagonalizable if there exists an invertible matrix P such that P-1AP is a diagonal matrix. Notes: (P diagonalizes
Diagonalization Math 240 Change of Basis Diagonalization Uses for diagonalization The change of basis matrix De nition Suppose V is a vector space with two bases B fv 1; 2;:::; n gand C w 1 2 n: The change of basis matrix from Bto Cis the matrix S [s ij], where v j s 1jw 1 s 2jw 2 s
Applications of Diagonalization Hsiu-Hau Lin hsiuhau@phys.nthu.edu.tw (Apr 12, 2010) The notes cover applications of matrix diagonalization (Boas 3.12). Quadratic curves Consider the quadratic curve, 5x2 4xy 2y2 30: (1) It can be casted into the matrix form and then brought into diagonal
Diagonalization Techniques for Sparse Matrices Rowan W. Hale May 17, 2012 Abstract We discuss several diagonalization techniques that fall into cate-gories of exact or iterative and direct or stochastic. Our discussion of these techniques has an emphasis on the runtime and memory usage (and a
Joint diagonalization of square matrices is an important problem of numeric computation. Many applications make use of joint diagonalization techniques as their main algorithmic tool, for example c 2004 Andreas Zie
From this point of view, it is natural to look for a diagonalization basis, namely, a basis of eigenvectors (eigen modes) for FN. In this regard, the main conceptual difficulty comes from the fact that the diagonalization problem is
Methods - Fast Diagonalization Matrix-vector multiplies can be recast as (A B)!u BUAT which can be done in O(n3) operations instead of O(n4)(even better savings in 3D) Best of all though is the Fast Diagonalization Property C A B B A C 1 1(VT VT) Under certain conditions
Diagonalization of Symmetric Matrices Let A 2Rn n be a symmtric matrix. Thm 1. Any two real eigenvectors pertaining to two
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