3 votes
Accepted

Why does A-sI have a better condition number than A?

The identity matrix is an optimally conditioned matrix: $κ(𝕀)=1$. Moreover, condition numbers are independent of rescaling: $κ(λA)=κ(A)$ for $λ≠0$. What $A-s𝕀$ does is it biases $A$ towards an ...
Hyperplane's user avatar
  • 11.7k
2 votes
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Uniqueness of Hessenberg matrix in Cholesky factorization of Hankel matrix

I would assume the upper Hessenberg matrix here is unreduced, i.e. the subdiagonal entries are all nonzero. The first column of $T$ is of course $Te_1$, and so is determined by (2.4). If $T$ is ...
dummy's user avatar
  • 500
2 votes
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Solving $Ax=b$: Projection onto subspace with a canonical basis of largest error

Similar to the steepest descent method, this is yet another example of a 1D projection method. The only difference is that we change a single component of $x_k$ to update the solution and this ...
Algebraic Pavel's user avatar
1 vote
Accepted

finite steps to Hessenberg form and/or triangular form

If you could reduce to a triangular matrix $A = QTQ^*$ (a Schur factorization) in a finite number of steps (involving elementary arithmetic operations and n-th roots only), this would violate the Abel-...
SGJ's user avatar
  • 367
1 vote
Accepted

Recommendations on numerical methods and numerical analysis books for machine learning?

Justin Solomon “Numerical Algorithms - Methods for Computer Vision, Machine Learning, and Graphics” (2015)
Georg Essl's user avatar
1 vote

Minimum columns of a matrix such that there is at least one nozero value in each row

This looks like the set cover problem. We look for a covering of $S:=\{1,\ldots,m\}$ by sets $S_j:=\{i: a_{ij}=1\}$, where $A=(a_{ij})$ is the given binary matrix. Since the problem is NP-hard, there ...
Algebraic Pavel's user avatar
1 vote
Accepted

Distance between subspaces with spectral norm

It is true that it could be made more explicit. Perhaps it would be more clear if the expression for $d(S_1,S_2)$ contained one important equality, namely $$ \left\|\begin{bmatrix}0&W_1^TZ_2\\-W_2^...
Algebraic Pavel's user avatar
1 vote

what is wrong with my equivalent transformation on optimization below

Note that if $A$ and $B$ are positive definite, then $A+B$ is positive definite. Suppose $v$ is an eigenvector corresponding to eigenvalue $\lambda$ of $B^{-1}A$, $B^{-1}Av = \lambda v$, then we have $...
Siong Thye Goh's user avatar
1 vote

Curve Fitting and continuous identification of parameterized matrix eigenvector from unconnected data points

One can use the following definition of an avoided crossing (anticrossing) between neighboring, ordered, eigenvalues at $s=s_c$: $$\lambda'_{k+1}(s_c)-\lambda'_{k}(s_c)=0, ~~~\lambda''_{k+1}(s_c)-\...
DinosaurEgg's user avatar
  • 10.8k
1 vote

Let $k$ and $w$ be digits and let $X$ be some positive integer with one or more digits. Using the two digits, $kw_7$ is a two digit base 7...

The base 7 and 9 stuff in the question is just another way of saying $$ 7k + w = 9w + k $$ in base 10 (although it doesn't really matter at this stage). Simplify, therefore $$ 3k = 4w $$ and the ...
Isco's user avatar
  • 93

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