Sylvester’s criterion – Knowledge and References

Sylvester's criterion

Sylvester's criterion is a mathematical test used to determine the definiteness of a matrix M. It is a necessary and sufficient condition for a symmetric matrix to be positive definite, meaning that all of its eigenvalues are positive. The criterion states that for a matrix to be positive definite, the determinants of all of its successive principal minors (which involve the major diagonal) must be positive. In other words, if all of the smaller submatrices formed by removing rows and columns from the original matrix have positive determinants, then the original matrix is positive definite.From: Modern Digital Control Systems [2019], Optimization in Food Engineering [2019]

Linear Discrete-Time Optimal Control

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Published in Raymond G. Jacquot, Modern Digital Control Systems, 2019

Raymond G. Jacquot

The Sylvester criterion is a test for the definiteness of a matrix M. It states that for a matrix to be positive definite, the minor determinants of all orders involving the major diagonal must all be positive. For a matrix M defined by () M = [m11m12⋯m1nm21m22⋯⋮mn1mnn]

Perfect tracking control of discrete-time quadratic TS fuzzy systems via feedback linearisation

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Published in International Journal of Systems Science, 2019

Xiaojun Ban, Liwei Ren, Zhibin Yan, Hao Ying

From Equation (13), we have The right side of the equation can be expressed in a matrix form , where and with Let , where is defined in Equations (14)–(16), with . Hence, Because of Equations (3) and (4) ( and ), it is obvious that at least one of is not zero, and the same for . It should be clear that , which consists of , are nonzero column vectors. Matrix is composed by with the rest of its elements being 0. Hence all column vectors of are linearly independent. Therefore, is a nonzero column vector. Note that is a symmetrical matrix. Thus, is in quadratic form. According to Sylvester’s criterion, , (or ) if and only if is positive definite (or negative definite).

Sylvester's criterion

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Linear Discrete-Time Optimal Control

Perfect tracking control of discrete-time quadratic TS fuzzy systems via feedback linearisation