Questions tagged [orthogonality]

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Finding the Vector $v$ for a Given Householder Matrix Transformation of Non-Collinear Vectors $a$ and $b$

Consider a vector $v$ in $\mathbb{R}^{n\times1}$. The Householder matrix is defined as follows: $$H(v)=I-\dfrac{2vv^T}{v^Tv}.$$ It can be demonstrated that $H(v)$ is symmetric and orthonormal. The ...
Ilkay Burak's user avatar
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Compute orthogonal complement using BLAS / LAPACK

Is there a fast method to compute an orthogonal complement of an arbitrary matrix $U\in\mathbb{R}^{m \times n}$ in BLAS / LAPACK? Specifically, I want any matrix $V\in \mathbb{R}^{m \times (m - \text{...
Bananach's user avatar
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Efficient and stable QR factorization of partially orthonormal matrix

Let $U \in \mathbb{C}^{m \times n_U}$ be an orthonormal matrix, let $A \in \mathbb{C}^{m \times n_A}$, and $m \geq n_U + n_A$. I want to compute a QR factorization $X = \left[U A\right] = QR$, with $Q ...
coolguy1000000's user avatar