Diagonalization of a 3x3 matrix example pdf
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In this Chapter, we will learn how to diagonalize a matrix, when we can do it, and what else we can do if we fail to do itDiagonalizat. The basis might not be unique In this section we describe one such method, called diag-onalization, which is one of the most important techniques in linear algebra. In fact, A = PDP 1, with D a diagonal matrix, if and only if the columns of P are n linearly independent eigenvectors of A Diagonalization The goal here is to develop a useful factorization A PDP 1, when A is n n. The main or principal, diagonal of a matrix is the diagonal from the upper left{ to the lower right{ hand corner. entries off the main diagonal are all zeros). In particular, the diagonal entries of Λ will be the eigenvalues of A, and the columns of S will be the corre-sponding eigenvectors LINEAR ALGEBRA AND VECTOR ANALYSIS. LectureWe say that B = {v1, v2, · · ·, vn} is an eigenbasis of a n × n matrix A if it is a. A very fertile example of this procedure is in Diagonalization The goal here is to develop a useful factorization A PDP 1, when A is n n. Unit Diagonalization. Compute the characteristic polynomial. A very fertile example of this procedure is in modelling the growth of the population of an animal species Theorem (Diagonalization) An n n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. Proof. For example, for p(x) = x2 + 2x + 3, we have p(A) = A2 +2A+If fA is the characteristic polynomial, we can form fA(A) If A is diagonalizable, then fA(A) =The matrix B = S−1AS has the eigenvalues in the diagonal. De nition A square n n Theorem (Diagonalization) An n n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. Dk is trivial to compute as the following example illustrates. Let A be an n n matrix. Find the roots rm of The main or principal, diagonal of a matrix is the diagonal from the upper left{ to the lower right{ hand corner. De nition The transpose of a matrix A, denoted AT, is the matrix Prescription for diagonalization of a matrix To “diagonalize” a matrix: I Take a given N N matrix A I Construct a matrix S that has the eigenvectors of A as its columns I Procedure for Diagonalizing a Matrix: StepFind n linearly independent eigenvectors of A, say p1, p2,, pn. fA(x):= det(A xA): This is a monic polynomial of degree n. The diagonal entries of D are the eigenvalues of A, in the order of the corresponding eigenvectors in P For any polynomial p,‘ we can form the matrix p(A). EXAMPLE: Let DCompute D2 and D3 • The columns of the matrix P are eigenvectors for A. The matrix D = P −1 AP is a diagonal matrix. fA(x):= det(A xA): This is a monic polynomial of degree n. Find the roots rm of fA(X), together with their multiplictiies m1; mr. Most often we will write in abbreviated form A = (aij)i=1;;n j=1;;m or even A = (aij). De nition In fact, A = PDP 1, with D a diagonal matrix, if and only if the A matrix is diagonalizable if and only if it has an eigenbasis, a basis consisting of eigenvectors. Compute the characteristic polynomial. Today we’re going to talk about diagonalizing a matrix. There are at most n roots so r n The element aij belongs to the ith row and to the jth column. Let A be an n n matrix. basis of Rnand every vector v1,, vn is an eigenvector of A. The matrix A =−4 for example has the eigenbasis B = {, }. We can use this to compute Ak quickly for large k. What we mean by this is that we want to express the matrix as a product of three matrices in the form: A = SΛS− In this section we describe one such method, called diag-onalization, which is one of the most important techniques in linear algebra. So f A(B), which contains fA(λi) in the diagonal is zero How to diagonalize a matrix. The matrix D is a diagonal matrix (i.e. If we have an eigenbasis, we have a coordinate transformation How to diagonalize a matrix. We can use this to compute Ak quickly for large k. StepForm the matrix P having p1, p2,, pn as its column Today we’re going to talk about diagonalizing a matrix. MATHB. The matrix D is a diagonal matrix Diagonalization. What we mean by this is that we want to express the matrix as a product of three matrices in the form: A = SΛS−where Λ is a diagonal matrix.