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Unformatted text preview: Eigenvalues, eigenvectors, eigenspaces, diagonalization, change of basis, and similarity I know, that’s a long title. But all these things are very intimately related, and it’s important that you understand how. Eigenstuff is very, very useful and will come up over and over again if you keep studying math, engineering or the physical sciences. So make sure you get it. Before relating all of these ideas, you need to know the basics of each. Let’s start with change of basis. Change of basis: Sometimes you have two different bases, and you want to change from one to the other – i.e., you want to take a vector expressed in coordinates with respect to one basis, and write it in coordinates with respect to another basis. The trick is to write one basis in terms of another, and put those coordinates as column vectors in a matrix. This matrix we’ll call the change of basis matrix. Let’s do this in more detail: Suppose S = { s 1 ,...,s n } and T = { t 1 ,...,t n } are two bases of a vector space V . In order to get the change of basis matrix from S to T , we first write: s 1 = c 1 1 t 1 + c 2 1 t 2 + ··· + c n 1 t n s 2 = c 1 2 t 1 + c 2 2 t 2 + ··· + c n 2 t n . . . s n = c 1 n t 1 + c 2 n t 2 + ··· + c n n t n Throw these coefficients into a matrix as columns like so: P = c 1 1 c 1 2 ··· c 1 n c 2 1 c 2 2 ··· c 2 n . . . . . . c n 1 c n 2 ··· c n n Then P will be our change of basis matrix from S to T . If we have a vector expressed in coordinates with respect to S , say, v = a 1 a 2 . . . a n (which means v = a 1 s 1 + a 2 s 2 + ··· + a n v n – remember, the a i are the coordinates of v with respect to S ), then we can multiply P a 1 a 2 . . . a n = b 1 b 2 . . . b n to get another list of 1 coordinates. This new list of coordinates, b 1 b 2 . . . b n , gives us the coordinates of v with respect to the basis T . The change of basis matrix from T back to S is given by P 1 . Let’s do a quick example. Let V = R 2 , and let S = { 1 1 , 1 1 } = { s 1 ,s 2 } and T = { 1 , 2 1 } = { t 1 ,t 2 } . Then we can find that s 1 = 1 / 2 t 1 1 / 2 t 2 and s 2 = 3 / 2 t 1 1 / 2 t 2 by solving a few linear systems. We can then get the change of basis matrix from S to T , P = 1 / 2 3 / 2 1 / 2 1 / 2 . If we have a vector, say, v = 3 s 1 2 s 2 then we can change from S coordinates (where v = 3 2 to T coordinates by multiplying by P , so the Tcoordinates of v are P 3 2 = 9 / 2 1 / 2 , i.e., v = 9 / 2 t 1 1 / 2 t 2 . Eigenvalues, eigenvectors and eigenspaces: Given a linear transfor mation L : V → V , if L ( v ) = λv for some v ∈ V , then we say that v is an eigenvector of L with eigenvalue λ . Often, linear transformations are described by giving a matrix that represents that linear transformation with respect to a given basis. That is, we are given some matrixgiven basis....
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This note was uploaded on 03/23/2010 for the course MAT 022A taught by Professor Pon during the Spring '10 term at UC Davis.
 Spring '10
 Pon
 Eigenvectors, Vectors

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