Which property is necessary for eigenvalues of a positive definite matrix?

The eigenvalues of a positive definite matrix must all be positive, which is crucial in understanding the characteristics of such matrices. A positive definite matrix ( A ) has the defining property that for any non-zero vector ( x ), the quadratic form ( x^T A x > 0 ). This property directly indicates that all eigenvalues are greater than zero.

When analyzing the implications of eigenvalues in the context of positive definite matrices, it is useful to consider the following: If a matrix is positive definite, it means that not only is it symmetric, but also that its quadratic form always yields a positive value for any non-zero vector. The positivity of the eigenvalues ensures that when we compute the corresponding quadratic forms, they maintain this positive nature.

In contrast, eigenvalues that are negative, zero, or imaginary would contradict the definition of positive definiteness. For example, a matrix with one or more negative eigenvalues could produce a negative quadratic form for certain vectors, violating the required condition for positive definiteness.

Therefore, the necessity for all eigenvalues to be positive is a fundamental characteristic of positive definite matrices, which ensures their suitability in various applications, including optimization problems and stability analysis in systems.