12/14/2023 0 Comments Orthogonal matrixThis means that cannot be real, as the nonreal eigenvalues of a real matrix must appear in complex conjugate pairs. We note first that the eigenvalues of are the square roots of those of and so of them will be and one will be. What can we say about square roots of a Householder matrix, that is, matrices such that ? In this case premultiplying a vector by flips the sign of the th component. If we take, the th unit vector, then, which has in the position. Then, so the component of in the direction has been reversed. Another way of expressing this property is to write, where is orthogonal to. This equation shows that reflects about the hyperplane, as illustrated in the following diagram, which explains why is sometimes called a Householder reflector. Note that the matrix is times a Hadamard matrix. Simple examples of Householder matrices are obtained by choosing, for which. Has trace and determinant, as can be derived directly or deduced from the facts that the trace is the sum of the eigenvalues and the determinant is the product of the eigenvalues.įor, a Householder matrix can be written as has eigenvalues with eigenvectors any set of linearly independent vectors orthogonal to, which can be taken to be mutually orthogonal: for every such.has an eigenvalue with eigenvector, since.The eigensystem can be fully described as follows. Where the last property follows from the first two.Ī Householder matrix is a rank- perturbation of the identity matrix and so all but one of its eigenvalues are. involutory ( that is, is a square root of the identity matrix),.A Householder matrix is an orthogonal matrix of the form
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