Math 2.0

statement theorem

QR decomposition

Statement

Let AA be an n×nn \times n complex matrix. Then there are a unitary matrix QQ and an upper triangular matrix RR with A=QR.A = QR. If AA is real, then QQ and RR can be chosen real, with QQ orthogonal. If AA is invertible and the diagonal entries of RR are required to be positive, then QQ and RR are unique.

Remarks

For an invertible AA, the Gram–Schmidt process applied to the columns of AA gives the columns of QQ. A rectangular m×nm \times n matrix with m≥nm \ge n has a decomposition A=QRA = QR with QQ unitary m×mm \times m and RR upper triangular m×nm \times n. Householder reflections and Givens rotations compute it stably. It is used to solve least squares problems and in the QR algorithm for eigenvalues.

Explanations

L1 human

The QR decomposition writes a matrix as a rotation, possibly with reflections, times a triangular matrix. The columns of QQ are orthonormal, and RR records how to rebuild the columns of AA from them. It is the matrix form of the Gram–Schmidt process.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionUpper triangular matrix, definition, no evidence levelUpper triangular matr…definitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionGram–Schmidt process, concept, no evidence levelGram–Schmidt processconceptQR decomposition, statement, no evidence levelQR decompositionstatement
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  1. Identity matrix object
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  9. Upper triangular matrix definition
  10. Conjugate transpose definition
  11. Orthogonal vectors definition
  12. Linear independence definition
  13. Linear span definition
  14. Unitary matrix definition
  15. Orthonormal family definition
  16. Gram–Schmidt process concept
  17. QR decomposition statement

Sources

  1. Web https://en.wikipedia.org/wiki/QR_decomposition, Square matrix

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