Tagged Questions

Questions on the various algorithms used in linear algebra computations (matrix computations).

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From $Ax=\lambda x$, we have $Ax i = \lambda x i$ , where $i^2=-1$??

Actually I found this problem when I met a question, asking me to prove the eigenvector and eigenvalue of real symmetric matrices are all real. I have already proved the eigenvalue part already, but ...
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Inverted pendelum Matrix numerical derivative

Here I've written a dynamic function as : ...
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Proof that $||f|| = \sqrt{\sum_{k=0}^{n}p(x_k)f(x_k)^2}$ is normed vector space

I've to prof given $X = \{x_1, x_2, ..., x_n\}$ and function $p$ with property $p(x: X) > 0$ that equation $||f|| = \sqrt{\sum_{k=0}^{n}p(x_k)f(x_k)^2}$ is norm on discrete set $X$. This ...
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How to get the SVD of $2AA^T-\operatorname{diag}(AA^T)$ given $A$ and its SVD $A=USV^T$?

Given a matrix $A\in R^{n\times d}$ with $n>d$, and we can have some fast ways to (approximately) calculate the SVD (Singular Value Decomposition) of $A$, saying $A=USV^T$ and $V\in R^{d\times d}$. ...
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roots of Padé approximating polynomials to the exponential function

I need to obtain (numerically) the roots of the denominator in the Padé approximation to the exponential function $e^{-x}$, in Python. I can calculate its coefficients in closed form (see below). But ...
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Perturbation of a linear homogeneous equation system

Let $A$ be a $n\times(n+1)$ matrix, full row rank. Let $\tilde A=A+\Delta A$ be a perturbation of $A$, again with full row rank. I am interested what is known about bounds on the angle between the ...
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Existence of Non-Commutative $4 \times 4$ Matrix Multiplication Algorithm

This paper by a Russian gentleman gives an optimal (?) algorithm for $3$ $\times$ $3$ matrix multiplication. It beats a previously known method by reducing the total number of discrete operations from ...
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Splitting Method

Consider the iteration matrix for the general splitting method $M=I-N^{-1}A$ where $N$ is any invertible matrix. Show that if $\lambda =1$ is an eigenvalue of $M$. then $A$ cannot be invertible. I ...
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Change multiple positions of points on circles with different radius

There are some points which are placed on a circular path: Now I want to change the position of some points equals to the distance value(d) respected to their path. I'm using this formula to ...
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