# Tag Info

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 ...
• 11.7k
Accepted

### 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 ...
• 500
Accepted

### 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 ...
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-...
• 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)
• 171
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 ...
1 vote
Accepted

• 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 ...
• 93

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