Geodesic
A characterization of the definiteness of a Hermitian matrix
Glasgow mathematical journal, Tome 15 (1974) no. 1, pp. 1-4

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We denote by F the field R of real numbers, the field C of complex numbers or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian (unitary resp.) if A = A* (AA*= identity matrix resp.). An n ×x n hermitian matrix A is said to be definite (semidefinite resp.) if uAu*vAv* ≥ 0 (uAu*vAv* ≧ 0 resp.) for all nonzero u and v in Fn. If A and B are n × n hermitian matrices, then we say that A and B can be diagonalized simultaneously into blocks of size less than or equal to m (abbreviated to d. s. ≧ m) if there exists a nonsingular matrix U with elements in F such that UAU* = diag{A1,..., Ak} and UBU* = diag{B1..., Bk}, where, for each i = 1, ..., k, Ai and Bk are of the same size and the size is ≧ m. In particular, if m = 1, then we say A and B can be diagonalized simultaneously (abbreviated to d. s.). 
Au-Yeung, Yik-Hoi; Yuen, Tai-Kwok. A characterization of the definiteness of a Hermitian matrix. Glasgow mathematical journal, Tome 15 (1974) no. 1, pp. 1-4. doi: 10.1017/S0017089500002019
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