1、Diagonalization and Eigendecomposition5th week / Linear AlgebraObjectives of This Week2The goal is to understandDiagonalizationEigendecompositionLinear transformation via eigendecomposition3Diagonalization We want to change a given square matrix into a diagonal matrix via the following form: = 1wher
2、e is an invertible matrix and is a diagonal matrix. This is called a diagonalization of . It is not always possible to diagonalize . For to be diagonalizable, an invertible should exist such that 1 becomes a diagonal matrix. 4Finding and How can we find an invertible and the resulting diagonal matri
3、x = 1? = 1 = Let us represent the following: = 1 2 where s are column vectors of =1 0 00 2 00 0 5Finding and = 1 2 = 1 2 = 1 2 1 0 00 2 00 0 = 11 22 = 1 2 = 11 22 6Finding and Equating columns, we obtain1 = 11, 2 = 22, = Thus, 1, 2, , should be eigenvectors and 1, 2, , should be eigenvalues. Then, F
4、or = = 1 to be true, should invertible. In this case, the resulting diagonal matrix has eigenvalues as diagonal entries. 7Diagonalizable Matrix For to be invertible, should be a square matrix in , and should have linearly independent columns. Recall columns of are eigenvectors. Hence, should have li
5、nearly independent eigenvectors. It is not always the case, but if it is, is diagonalizable. 8Eigendecomposition If is diagonalizable, we can write = 1. We can also write = 1. which we call eigendecomposition of . being diagonalizable is equivalent to having eigendecomposition. 9Linear Transformatio
6、n via Eigendecomposition Suppose is diagonalizable, thus having eigendecomposition Consider the linear transformation x = x. x = x = 1x = 1x . = 110Change of Basis Suppose 1 = 11 and 2 = 22. x = x = 1x = 1x Let y = 1x. Then, y = x y is a new coordinate of x with respect to a new basis of eigenvector
7、s 1, 2 . x = 43 = 4 10 +3 01 = 10 01 43 = y = 1 2 12 = 21 +12 y = 21x = 43 1211Element-wise Scaling x = 1x = y Let = y. This computation is a simple Element-wise scaling of y. Example: Suppose = 1 00 2 . Then = y = 1 00 2 21 = 1 221 = 2212Dimension-wise Scaling = 22 12y = 21 1213Back to Original Bas
8、is x = y = is still a coordinate based on the new basis 1, 2 . converts to another coordinates based on the original standard basis. That is, is a linear combination of 1 and 2using the coefficient vector . That is, = 1 2 12 = 11 +22 x = = 1 2 22= 21 +22= 2 31 +2 21= 10014Back to Original Basis = 22
9、1215Overview of Transformation using Eigendecomposition = = 1 16Linear Transformation via Now, consider recursive transformation = . If is diagonalizable, has eigendecomposition = 1 = 1 1 1 = 1 is simply computed as =1 0 00 2 00 0 17Linear Transformation via = 1 can be computed in the similar manner to the previous example. It is much faster to compute 1xthan to compute . DiagonalizationEigendecompositionLinear transformation via eigendecompositionSummary
