Quadratic Forms
The (purely) quadratic form associated with a real matrix M is the function Q from to such that for all x. It turns out that the matrix M is positive definite if and only if it is symmetric and its quadratic form is a strictly convex function.
More generally, any quadratic function from to can be written as where is a symmetric n×n matrix, b is a real n-vector, and c a real constant. This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only if M is positive definite. For this reason, positive definite matrices play an important role in optimization problems.
Read more about this topic: Positive-definite Matrix
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“A painter told me that nobody could draw a tree without in some sort becoming a tree; or draw a child by studying the outlines of its forms merely,—but by watching for a time his motions and plays, the painter enters into his nature and can then draw him at will in every attitude.”
—Ralph Waldo Emerson (1803–1882)