You are implementing a closed-form ridge regression solution for a data set with 20 000 observations (rows) and 300 predictors (columns). The algorithm requires the Gram matrix
G = X^T X (shape 300 × 300)
where X is the design matrix. A teammate suggests computing
H = X XT (shape 20 000 × 20 000) and then taking HT, claiming that the transpose will convert H into G while avoiding an extra allocation. Which statement correctly evaluates this suggestion?
It works only when X has orthonormal columns, since then X XT and XT X are identical up to shape.
The approach succeeds by leveraging the Woodbury identity; transposing H is merely an optimization step.
It will work because the transpose reverses multiplication order, so (X X^T)T becomes XT X with the desired 300 × 300 shape.
It will fail; (X XT)T equals X XT, so the result is still 20 000 × 20 000 and cannot substitute for XT X unless X is square.
The transpose of a product reverses the order of the factors: (AB)T = BT AT. Applying this rule to H gives (X XT)T = (XT)T XT = X XT, so the matrix is unchanged-not converted into XT X. Because X has many more rows than columns, H is 20 000 × 20 000 whereas G is 300 × 300; the dimensions alone show that transposing H cannot yield G unless X were square. Therefore the proposal fails. The other options misunderstand the transpose rule or apply conditions (orthonormal columns, Woodbury identity, full rank) that do not change the dimension mismatch.
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