| Householder Transformation |
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| CATEGORIES ABOUT HOUSEHOLDER TRANSFORMATION | |
| geometry | |
| linear algebra | |
| matrices | |
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The Householder transformation was introduced in 1958 by Alston Scott Householder . It can be used to obtain a QR Decomposition of a matrix. DEFINITION AND PROPERTIES The reflection hyperplane can be defined by a Unit Vector (a vector with length 1), that is Orthogonal to the hyperplane. If is given as a column unit vector and is the Identity Matrix the linear transformation described above is given by the Householder matrix ( denotes the Transpose of the vector ) : . The Householder matrix has the following properties:
Furthermore, really reflects a point X (which we will identify with its position vector ) as described above, since : , where denotes the Dot Product . Note that is equal to the distance of X to the hyperplane. APPLICATION: QR DECOMPOSITION Householder reflections can be used to calculate QR decompositions by reflecting first one column of a matrix onto a multiple of a standard basis vector, calculating the transformation matrix, multiplying it with the original matrix and then recursing down the (''i'',''i'') Minor s of that product. See the QR Decomposition article for more. REFERENCES
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