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Tham khảo tài liệu 'applied structural and mechanical vibrations 2009 part 17', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Theorem A.5. If A đụ- Ẽ M X has eigenvalues Al At.A the following statements are equivalent 1. A is normal. 2. A is unitarily diagonalizable. 3. gZ Ị đ I2 I A. I 4. There is a orthonormal set of n eigenvectors of A. The equivalence of 1 and 2 in Theorem A.5 is often called the spectral theorem for normal matrices. For our present purposes we recall that a Hermitian matrix is just a special case of normal matrix and we stress that as expected the statement of the theorem says nothing about A having distinct eigenvalues and in fact two or more eigenvalues could be equal . Then summarizing the results of the preceding discussion we can say that a complex Hermitian matrix or a real symmetrical matrix A 1. has real eigenvalues 2. is always nondefective which means that regardless of the existence of multiple eigenvalues there always exists a set of n linearly independent eigenvectors which in addition are mutually orthogonal 3. is unitarily orthogonally similar to the diagonal matrix of eigenvalues diagf j . Moreover the unitary orthogonal similarity matrix is the matrix X of eigenvectors in which the th column is the th eigenvector. We close this section by briefly considering special classes of Hermitian matrices. A nxn Hermitian matrix A is said to be positive definite if xHAx 0 A.32a for all nonzero vectors X e -Ể . If the strict equality in eq A.32a is weakened to xhAx 0 A.32b then A is said to be positive semidefinite. Moreover by simply reversing the inequalities in eqs A.32a and A.32b we can define the concept of negative definite and negative semidefinite matrices. Note that if A is Hermitian the definitions above tacitly imply that the term xHAx which is called the Hermitian form generated by A is always a real number and so we can also speak of positive definite Hermitian form eq A.32a or positive semidefinite Hermitian form eq A.32b . The real counterparts of Hermitian forms are called quadratic forms and are expressions of the type xTAx where A is a real .
