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- Thread starter golmschenk
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[tex]f(x) = \frac{1}{2} x A^T x + b^Tx + c [/tex]

has a unique minimum (expressions like these crop up in a number of places). A positive definite matrix A can be visualized as a paraboloid (look at the graph of f) that is stretched in the directions of A's eigenvectors. If A is indefinite, the graph will have a saddle point instead of a nice minimum (or be degenerated further).

An article that explains this (and some other linear algebra key ideas) nicely is "Introduction to the Conjugate Gradient Method Without the Agonizing Pain" by JR Shewchuck.

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AlephZero

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This means there are usually faster and simpler numerical algorithms for positive definite matrices than for general matrices.

In physics, matrices are often Hermitian (which includes real symmetric matrices) as well as positive definite, and the product x^t A x represents some kind of work or energy.

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