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More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×m unitary matrix Q and an m×n upper triangular matrix R.As the bottom (m−n) rows of an m×n upper triangular matrix consist entirely of zeroes, it is often useful to partition R, or both R and Q:
An RRQR factorization or rank-revealing QR factorization is a matrix decomposition algorithm based on the QR factorization which can be used to determine the rank of a matrix. [1] The singular value decomposition can be used to generate an RRQR, but it is not an efficient method to do so. [2] An RRQR implementation is available in MATLAB. [3]
Instead, the QR algorithm works with a complete basis of vectors, using QR decomposition to renormalize (and orthogonalize). For a symmetric matrix A , upon convergence, AQ = QΛ , where Λ is the diagonal matrix of eigenvalues to which A converged, and where Q is a composite of all the orthogonal similarity transforms required to get there.
QR decomposition, a decomposition of a matrix QR algorithm, an eigenvalue algorithm to perform QR decomposition; Quadratic reciprocity, a theorem from modular arithmetic; Quasireversibility, a property of some queues; Reaction quotient (Q r), a function of the activities or concentrations of the chemical species involved in a chemical reaction
In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization).
At the very beginning of the page, there is a mistake: "In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R." I think it's worth noting that Q is only orthogonal if it's square.
For the QR algorithm with a reasonable target precision, this is , whereas for divide-and-conquer it is . The reason for this improvement is that in divide-and-conquer, the Θ ( m 3 ) {\displaystyle \Theta (m^{3})} part of the algorithm (multiplying Q {\displaystyle Q} matrices) is separate from the iteration, whereas in QR, this must occur in ...
In Learning the parts of objects by non-negative matrix factorization Lee and Seung [43] proposed NMF mainly for parts-based decomposition of images. It compares NMF to vector quantization and principal component analysis , and shows that although the three techniques may be written as factorizations, they implement different constraints and ...