Cochran's theorem

Cochran's theorem characterizes when quadratic forms of a multivariate normal vector are independent and have scaled chi-squared distributions, using conditions on their matrices and ranks.

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Quadratic form: A scalar expression formed by multiplying a vector, a matrix, and the vector’s transpose. The theorem’s random variables are quadratic forms in a normal vector.
Independence (probability theory): A relationship in which knowledge of one random event or variable does not change the probabilities of another. Cochran’s theorem turns algebraic orthogonality conditions into independence of quadratic forms.
Analysis of variance: A statistical method that partitions observed variation into components associated with specified sources. Its sums of squares are independent chi-squared components under the normal-model assumptions.
Craig's theorem: A theorem giving a matrix condition for independence between quadratic and linear forms of a multivariate normal vector. It treats quadratic and linear forms, whereas Cochran’s theorem focuses on collections of quadratic forms.
Idempotent matrix: A square matrix whose square equals itself. Projection matrices in the theorem are idempotent, linking their ranks to chi-squared degrees of freedom.
Rank (linear algebra): The dimension of the column space of a matrix. The ranks of the relevant projection matrices give the chi-squared degrees of freedom.
Linear regression: A model expressing a response as a linear function of predictors plus an error term. Regression residual and fitted sums of squares follow Cochran-style decompositions under Gaussian errors.
Basu's theorem: A theorem stating that an ancillary statistic is independent of a complete sufficient statistic. It provides a different route to independence in statistics, based on sufficiency rather than projection matrices.
Orthogonal projection: A linear map that projects vectors onto a subspace perpendicular to its residual space. Orthogonal projections generate the independent quadratic forms in the theorem’s standard decomposition.
Positive semidefinite matrix: A symmetric matrix whose quadratic form is nonnegative for every real vector. The theorem’s standard quadratic-form matrices are symmetric projections, hence positive semidefinite.