Positive-definite matrices

A symmetric or Hermitian matrix is positive definite when its quadratic form is positive on nonzero vectors.

Different ways to understand it

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Matrix norms
Matrix norms quantify matrix size; induced norms measure maximum vector amplification.
Orthogonal complements
Vectors orthogonal to every vector of a subspace form its orthogonal complement.
Singular values
Singular values are nonnegative square roots of eigenvalues of the conjugate-transpose product.
Tensor products
Tensor products linearize bilinear maps and combine vector spaces.
Basis and dimension
A basis is a linearly independent spanning set; every vector has unique coordinates in that basis.