QR Decomposition
The problem, from first principles
12-gram-schmidt turns a set of vectors into an orthonormal basis, one vector at a time, discarding the "how much of each original vector was removed" bookkeeping along the way. QR decomposition is Gram-Schmidt with that bookkeeping kept: the orthonormal basis becomes matrix Q, and exactly how each original column was built from it becomes matrix R — together, Q and R capture the same information as the original matrix, just split into an orthogonal part and a triangular part.
From theory to code
Implement qr_decompose(A), returning (Q, R) such that A = QR, built directly from 12-gram-schmidt's orthonormalization process applied column-by-column. The signature and docstring are already in the editor.
Constraints
Ais anm × nNumPy array whose columns are linearly independent (a requirement Gram-Schmidt itself already assumes).Q's columns are orthonormal (12-gram-schmidt's output).Ris upper-triangular.Q @ Rmust reconstructAexactly (up to floating-point precision).
Hints
Hint 1
Apply Gram-Schmidt to A's columns one at a time to get Q's columns. R's entries are exactly the projection coefficients Gram-Schmidt computes and subtracts along the way — record them instead of throwing them away.
Hint 2
R[i, j] (for i <= j) is the dot product of Q's i-th column with A's original j-th column; R[j, j] specifically is the norm of what's left of column j after subtracting off its projections onto every earlier Q column, before that leftover gets normalized into Q's j-th column.
Keywords
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