By Mac Lane, Birkhoff (WEIL, HOCQUEMILLER)

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**Additional info for Algebre, solutions developpees des exercices, 1ere partie, ensembles, groupes, anneaux, corps [Algebra]**

**Sample text**

Eigen values thus obtained need not be distinct. For every distinct Eigen values, there exist one or more Eigen vectors that are obtained as follows. The Eigen vectors corresponding to the Eigen value 1 are obtained as the basis of the Null space of the matrix ŒA 1 I . 20. 18 Eigen Values and Eigen Vectors 43 The Eigen values of the matrix A are given as 1 D1 p 7 C 57 D 7:2749 2 D 2 p 7 57 D 0:2749 3 D 2 The Eigen vectors corresponding to the distinct Eigen values are obtained as follows. Eigen vector corresponding to the Eigen value 1 D 1 is the null space of the matrix ŒA 2 3 1 0 0 1 I D A D 40 2 35 0 4 5 2 3 2 3 1 0 0 0 0 0 1 40 1 05 D 40 1 35 0 0 1 0 4 4 2 3 0 0 0 The Null space of the matrix 40 1 35 is obtained as follows.

B bc /. 1/ 55 b x2 6 b (b-bc) Fig. AT A/ 2 5 A 25 7 5 D Œb b x2 3 6 3 6 36 10 2 1 AT b which is same as the one used in the previous section. AT A/ 1 AT b AŒb x is the vector bc that lies in the column space of the matrix A. AT A/ 1 AT is called as the projection matrix, that projects the vector in the vector space R3 into the column space of the matrix A such that kbc bk is minimized. AT A/ D Identity matrix. AT A/ the left inverse of the matrix A, which is usually represented as AC . AT A/ 1 exists.

The DFT matrix is the unitary matrix. Four-point DFT matrix is as shown below. 3 2 1 1 1 1 Â Ã 2 37 1 6 61 w w w 7 AD 2 4 4 1 w w w6 5 2 1 w3 w6 w9 j2 Á where w D e 4 Note that the columns of the matrix A are orthonormal to each other. i:e/A0 0 1 0 0 0 0 1 0 3 0 07 7 05 1 1. The Eigen values of the matrix A are 1, 1, i 2. The Eigen vectors corresponding to the above mentioned Eigen values are given as 3 3 2 3 2 3 2 0:0231 0:0070i 0 0:5 0:8660 60:28877 6 0:5 7 6 0:7071 7 6 0:4158 0:0035i 7 7 7 6 7 6 6 7 6 5 40:28875 ; 4 0:5 5 ; 4 0 5 ; 4 0:8085 0:4158 0:0035i 0:7071 0:5 0:2887 2 Characteristics of the DFT Matrices 1.