Section36supplementTimeless - Econ 204 Supplement to...

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Econ 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms De La Fuente notes that, if an n × n matrix has n distinct eigenvalues, it can be diagonalized. In this supplement, we will provide an additional, very important diagonalization result. Symmetric matrices can always be diagonalized; moreover, the change of basis matrices that carry out the di- agonalization have a special form. This has an important application to quadratic forms, which in turn have application to the geometry of level sets of preferences, and to the analysis of variance-covariance matrices. 1 Diagonalization and Change of Basis Before proceeding with the diagonalization result for symmetric matrices, it is useful to discuss the relationship between diagonalization and change of basis. De La Fuente (page 151) de±nes a square matrix M to be diagonalizable if there exists an invertible matrix P such that P 1 MP is diagonal; he also (page 146) de±nes two square matrices A and B to be similar if there is an invertible matrix P such that P 1 AP = B ,soasqua r ema t r ix M is diagonalizable if and only if it is similar to a diagonal matrix. Theorem 2 tells us that a matrix is diagonalizable if and only if there is another basis so that the representation of the same transformation in the new basis is diagonal. Proposition 1 Fix an n -dimensional vector space X and a basis U = { u 1 ,...,u n } . An n × n matrix P is invertible if and only if there is a basis W such that P =( Mtx ) U,W ( id ) ; in this case, (( ) U,W ( id )) 1 ) W,U ( id ) . Proof: Suppose that P is invertible. Let W = { w 1 ,...,w n } ,whe re w j = n i =1 p ij u i .S i n c e P is invertible, rank P = n ,s o W is a basis. P = ( ) U,W ( id ). Conversely, suppose there is a basis W such that P ) U,W ( id ). Then w j = n i =1 p ij u i . By the Commutative Diagram Theorem (see the 1
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Supplement to Section 3.3), ( Mtx ) U,W ( id ) · ( ) W,U ( id )=( ) U,U ( id id ) =( ) U ( id ) = I so P is invertible and ( ) W,U ( id )=(( ) U,W ( id )) 1 . Theorem 2 Suppose that X is fnite-dimensional IF T L ( X, X ) ,and U, W are any two bases oF X ,then ( ) W ( T ) and ( ) U ( T ) are similar. Conversely, given similar matrices A, B with A = P 1 BP and any basis U , there is a basis W and T L ( X, X ) such that B ) U ( T ) , A ) W ( T ) , P ) U,W ( id ) P 1 ) W,U ( id ) .
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Section36supplementTimeless - Econ 204 Supplement to...

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