204Lecture102008Web - Economics 204 Lecture 10Friday,...

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Economics 204 Lecture 10–Friday, August 8, 2008 Diagonalization of Symmetric Real Matrices (from Handout): De±nition 1 Let δ ij = 1i f i = j 0i f i 6 = j Abas is V = { v 1 ,...,v n } of R n is orthonormal if v i · v j = δ ij . In other words, each basis element has unit length, and distinct basis elements are perpendicular. Observation : Suppose that x = n j =1 α j v j where { v 1 n } is an orthonormal basis of V . Then for any x V , x · v k = n X j =1 α j v j · v k = n X j =1 α j ( v j · v k ) = n X j =1 α j δ jk = α k so x = n X j =1 ( x · v j ) v j Example: The standard basis of R n is orthonormal. De±nition 2 Area l n × n matrix A is unitary if A > = A 1 ,where A > denotes the transpose of A :th e ( i, j ) th entry of A > is the ( j, i ) th entry of A . Theorem 3 l n × n matrix A is unitary if and only if the columns of A are orthonormal. 1
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Proof: Let α j denote the j th column of A . A > = A 1 A > A = I α i · α j = δ ij ⇔{ α 1 ,...,α n } is orthonormal If A is unitary, let V be the set of columns of A and W be the standard basis of R n . Since A is unitary, it is invertible, so V is a basis of R n . A > = A 1 = Mtx V,W ( id ) Since V is orthonormal, the transformation between bases W and V preserves all geometry, including lengths and angles. Theorem 4 Let T L ( R n , R n ) , W the standard basis of R n . Suppose that W ( T ) is symmetric. Then there is an orthonormal basis V = { v 1 ,...,v n } of R n consisting of eigenvectors of T , so that W ( T ) is diagonalizable: W ( T ) = W,V ( id ) · V ( T ) · ( id ) where V T is diagonal and the change of basis matrices ( id ) and W,V ( id ) are unitary. The proof of the theorem requires a lengthy digression into the linear algebra of complex vector spaces. Here is a very brief outline. 1. Let M = W ( T ). 2. The inner product in C n is deFned as follows: x · y = n X j =1 x j · y j 2
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where ¯ c denotes the complex conjugate of any c C ; note that this implies that x · y = y · x .Th e usual inner product in R n is the restriction of this inner product on C n to R n . 3. Given any complex matrix A , deFne A to be the matrix whose ( i, j ) th entry is a ji ; in other words, A is formed by taking the complex conjugate of each element of the transpose of A .I ti se a s yt o verify that given x, y C n and a complex n × n matrix A , Ax · y = x · A y .S i n c e M is real and symmetric, M = M . 4. If λ C is an eigenvalue of M , with eigenvector x C n ,then λ | x | 2 = λ ( x · x ) =( λx ) · x Mx ) · x = x · ( M x ) = x · ( ) = x · ( λx ) = ( λx ) · x = λ ( x · x ) = λ | x | 2 = ¯ λ | x | 2 which proves that λ = ¯ λ , hence λ R .
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.

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204Lecture102008Web - Economics 204 Lecture 10Friday,...

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