Articles containing proofs | Matrix theory

Sylvester's criterion

In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: * the upper left 1-by-1 corner of M, * the upper left 2-by-2 corner of M, * the upper left 3-by-3 corner of M, * * M itself. In other words, all of the leading principal minors must be positive. By using appropriate permutations of rows and columns of M, it can also be shown that the positivity of any nested sequence of n principal minors of M is equivalent to M being positive-definite. An analogous theorem holds for characterizing positive-semidefinite Hermitian matrices, except that it is no longer sufficient to consider only the leading principal minors:a Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative. (Wikipedia).

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From playlist Extreme Value Theorem of Functions

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From playlist IISc Bangalore: Ordinary Differential Equations and Applications | CosmoLearning.org Mathematics

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From playlist Extreme Value Theorem of Functions

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From playlist Extreme Value Theorem of Functions

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From playlist Latest talks and lectures

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From playlist Extreme Value Theorem of Functions

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From playlist Seminar In the Analysis and Methods of PDE (SIAM PDE)

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From playlist Extreme Value Theorem of Functions

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👉 Learn how to find the extreme values of a function using the extreme value theorem. The extreme values of a function are the points/intervals where the graph is decreasing, increasing, or has an inflection point. A theorem which guarantees the existence of the maximum and minimum points

From playlist Extreme Value Theorem of Functions

Related pages

Orthogonal matrix | Cholesky decomposition | Triangular matrix | Determinant | James Joseph Sylvester | Block matrix | Spectral theorem | Hermitian matrix | Mathematical induction | QR decomposition