4 votes
Accepted

The point in a plane that minimizes the sum of distances to points outside it

If $x_1$ and $x_2$ are on different sides of $V$, you want $x^*$ to be where the straight line from $x_1$ to $x_2$ intersects $V$. The reason that this is true should be rather obvious, but can be ...
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  • 188k
2 votes

Computing $(A^TA)A^T$ versus $A^T(AA^T)$ on a Computer with Limited Memory where $A$ is $1000000 \times 2$

In the context of someone who only knows about the definition of matrix multiplication, would this be correct? $A^TA$ is a $2 \times 2$ matrix (4 total cells), while $AA^T$ is a $1000000 \times ...
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  • 9,503
2 votes
Accepted

How unique is thin/reduced QR decomposition without $R_{ii}>0$ condition?

The sticking point in my proof is the assumption that the 𝑅 matrices are invertible (or simply that they have nonzero diagonals). Notice for any choice of $k$ $0\lt \det\big(A^T A\big) = \det\big(...
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  • 7,743
1 vote

Efficiently compute rows of matrix which are not positive combinations of other rows

$\newcommand{\R}{\mathrm{I\!R}}\newcommand{\cone}{\mathrm{cone}}$I will give an answer regarding the columns of $A$ (I find it more convenient). If you are interested in the rows of $A$ you can apply ...
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1 vote
Accepted

How do Krylov methods performance wise compare to direct methods when no preconditioning is done?

Poorly. The fundamental problem is that Krylov subspace methods have low arithmetic intensity and are incompatible with modern computer architecture In contrast, modern implementations of the QR and ...
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1 vote

Cholesky-like decomposition that works on singular matrices?

An approach using $QR$ decomposition: the following results in a decomposition of the form $A = MM^T$ with $M$ of full rank of size $n \times r$ (with $r$ equal to the rank of $A$), but $M$ is not ...
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1 vote

Computing $(A^TA)A^T$ versus $A^T(AA^T)$ on a Computer with Limited Memory where $A$ is $1000000 \times 2$

The product of an $m\times n$ and an $n \times k$ matrix is an $m \times k$ matrix, and each entry requires $n$ multiplications to compute, for a total of $mnk$ multiplications (assuming the naive ...
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  • 22.1k
1 vote

Gram matrix of QR decomposition

The projection $\hat x$ of $x$ onto the image of $A$ is $$ \hat x = A(A^T A)^{-1} A^T x. \tag{1} $$ so $y = x - 2 A(A^T A)^{-1} A^T x $. This is a well known formula in the topic "linear ...
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  • 5,848
1 vote
Accepted

Solution of Diffusion equation with exponential integrator

There are several options: Find a Similarity transform $S$ that diagonalizes $A$, and then $e^{A dt}=S^{-1} e^{Ddt}S$ with $D$ diagonal. Realize that in any case you are dealing with an discrete time ...
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  • 2,969
1 vote
Accepted

Arnoldi iteration and minimal poly

As I note in my comment, your approach is incorrect. The key is to note that Arnoldi iteration runs to completion with the initial vector $b$ if and only if the Krylov subspace generated by $b$ is ...
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