1 Finally, a sufficient and necessary condition of the diagonalizable block compound matrices is proved.

2 By decomposing a matrix into one diagonalizable matrix and two orthogonal matrixes, singular value decomposition has very good properties.

3 This paper discusses the structure, calculation of multiplication and power, eigenvalue and eigenvector, and diagonalizable problems of matrix of rank equal to 1.

4 In the final analysis, it will turn out that a transformation is diagonalizable if the matrix associated with it is symmetric or Hermitian .

5 Lastly, the new relative perturbation bounds for special matrix and diagonalizable matrix are obtained.

6 One kind of inverse eigenvalue problems, whose solutions are required to be normal or diagonalizable matrices, is investigated in quaternionic quantum mechanics.

7 The main purpose of this paper is to study a new proof of extended unitary diagonalizable of self-conjugate matrix in quaternion field and deduce other properties concerned.