Math 220 Assignment

November 30, 2001

The Characteristic Equation

If, relative to a given coordinate system in an n-dimensional vector space, the columns of an invertible n \times n matrix A form a basis of that vector space relative to which a linear transformation that is represented in the given coordinate system by a matrix M is diagonalized, i.e., represented by a diagonal matrix D, then

 A^{-1} M A  =  D  .  

Equivalently M A = A D, and, taking the j^{th} column one sees that

 M A_{j}  =  (M A)_{j}  =  (A D)_{j}  =  A D_{j}  =  d_{jj} A_{j}  .  

Thus, each member A_{j} of the diagonalizing basis must lie in the kernel of the linear function represented in the given coordinate system by the matrix M - t 1_{n}, where 1_{n} denotes the n \times n identity matrix, when t = d_{jj}. Thus, each A_{j} may be found by finding the kernel of M - t 1_{n} when t = d_{jj}, and the diagonal elements d_{jj} of D may be found among the roots of the characteristic polynomial equation

 det
(M - t 1_{n})
  =  0  .  

Due Monday, December 3

  1. Find the characteristic polynomial and its roots for each of the matrices

    (
    3 
    4
    4
    -3
    		)
    		       and       
    (
    -1 
    0
    1
    1
    		)
    		  .  

  2. Let S be the 3 \times 3 matrix

     
    (
    10 
    -6
    -2
    -6
    5
    -8
    -2
    -8
    3
    		)
    		  .  
    Find an orthogonal matrix U and a diagonal matrix D such that
      S  =  U D U^{-1}  .  

AUTHOR  |  COMMENT