# Questions tagged [numerical-linear-algebra]

{numerical-linear-algebra} questions involving algorithms for linear algebra computations.

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### When can a point be reconstructed from relative angle measurements?

Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...
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### The selection of minimal generating sets in Lie algebra

Suppose $A$ is a Lie algebra on field $F_{p}$ with $[A,A,A]=0$. Denote $\{a_{1},\cdots,a_{d}\}$ is a minimal generating set of $A$.It's possible that $[a_{i},a_{j}]=0$ for some $1\leq i<j\leq d$ ...
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### Efficiently solve the Sylvester equation $AX+XA = C$ where $X$ is skew-symmetric

Is there a way (more efficient than the standard vectorization) to solve the following Sylvester equation in the skew-symmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive ...
• 173
1 vote
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### Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart

I am trying to understand the connection between the eigenspace of the continuous operator $$H(x,y) = \frac{1}{x+y}$$ which is nothing but the square of the Laplace operator, and its discrete ...
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### Backward stability of the SVD

I am interested in the backward stability of numerical algorithms for computation of the singular value decomposition (SVD). Specifically, I am interested in the following result: Backward stabile ...
• 283
1 vote
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### The geometrical multiplicity of the nilpotent matrices

The following point is well-known in the literature. Theorem. Let $A$ be a non-negative matrix in $M_n(\mathbb{R})$. If $A$ is nil-potent, there is a permutation matrix $P$ such that $P^tAP$ is ...
• 4,030
1 vote
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### Characterization of the behavior of the residuals in conjugate gradient

In conjugate gradient method for solving symmetric positive definite linear system $Ax=b$, which can also be regarded as a convex optimization problem $\dfrac{1}{2} x'Ax - x'b$, the $A$-norm of the ...
• 205
1 vote
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### QR algorithm for eigenvalues and eigenvectors of large symmetric matrices

I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices, My initial thought was to use Householder transformation with a Wilkinson shift ...
3k views

### Quickly determining if a matrix has any PSD completion

Given $m$ entries of an $n \times n$ matrix, is it possible to determine in $O(m n)$ time whether there is any positive semidefinite completion? Slightly more precisely: for simplicity let's assume ...
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### Complexity of inverting and multiplying against a symmetric Toeplitz matrix with two repeated entries

I know that the computational complexity of inverting a general $n \times n$ matrix $A$ is $O(n^{2.373})$ and multiplying it against an $n \times m$ matrix is $O(n^2m)$. Moreover, I've seen that ...
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### Optimal Truncation of LDL-factorization to improve conditioning

Suppose I factored real symmetric quasi-definite $A_0= L_0 \cdot D_0 \cdot L_0^T$ and the factorization exists, with $D$ diagonal and $L$ unit lower-triangular; and suppose $L$ and $D$ are badly ...
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1 vote
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### Slope assertion in Cholesky on digital computers

For a real symmetric positive definite linear system $$A \cdot x = b,$$ solved using Choelsky with forward- and backward-substitution, we know it for the numerical approximation $\tilde{x}$ to $x$ ...
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### Practical symmetric equivalent to QR factorization updates

As we know, the QR-factorization $Q\cdot R=A$ of any real symmetric $n \times n$ matrix $A$ with full rank is unconditionally numerically stable. Further, when A is rank-1-updated, the factorization ...
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### $\det(HH’) = 0$ for nonnegative $H$

$H$ is an $n\times m$ matrix with non-negative coefficients and $n < m$. $H'$ is the transpose of $H$. Are the following statements true? If $\det(HH’) > 0$, the rows of $H$ define the edges of ...
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1 vote