text: # Tag Info
  
  Accepted
  
  • 8,685
  Accepted
  
  ### Why is in Matlab exp(pi * sqrt(163)/3) - 640320 = -2.3283e-10
  
  First, that expression is not an integer, but very close to an integer $640320$. In fact, as @Peter mentioned, this is related to some deep theory of modular $j$-functions and Heegner numbers, e.g. ...
  • 15.2k
  Accepted
  
  ### What unique value does Cramer's rule offer?
  
  Cramer‘s rule is used in a lot of proofs. You can use it to get the $(i,j)$-th entry of the matrix $A^{-1}$ (see adjugate matrix), it be used to prove the that every smooth $C^1$ diffeomorphism is ...
  • 1,130
  Accepted
  
  ### Prove that $1 \leq \|A^{-1}\| _2\|A-B\|_2$
  
  Let $v$ be a nonzero unit vector in $\ker(B)$. Then $$\|A^{-1}\|_2\|A-B\|_2 \ge\|I-A^{-1}B\|_2 =\sup_{\|u\|_2=1}\|(I-A^{-1}B)u\|_2 \ge\|(I-A^{-1}B)v\|_2 =\|v\|_2=1.$$
  • 140k
  Accepted
  
  ### Why is this matrix always symmetric?
  
  An initial simplification helps. Absorb $\frac{h^2}{8}$ into $B$ and forget about it. Now note that since $B$ is symmetric, our given matrix is symmetric if and only if $X:=B(I-A^{-1}B)^{-1}$ is ...
  
  ### Intersection of two ellipses at exactly 2 points
  
  Assume that $A, B, k$ are given where $A, B$ are symmetric positive definite $2 \times 2$ matrices, and $k \gt 0$. The solution of $x^T A x = k$ is the ellipse $x = v_1 \cos t + v_2 \sin t$ ...
  • 23.4k
  
  ### Intersection of two ellipses at exactly 2 points
  
  We can assume WLOG (up to division by a constant) that the first ellipse $(E)$ has equation $$X^TAX=1$$ Let $X^TBX=\ell$ be the second variable ellipse that we will call $(E_{\ell})$. Result : the ...
  • 83.1k
  Accepted
  
  ### Why is easier to get inverse of mass matrix?
  
  The claim that "Inverting the mass matrix is significantly easier than solving a linear system involving the stiffness matrix" is more of a slogan than a precise theorem. This answer gives ...
  • 706
  
  • 12.7k
  
  ### Finding the bounds of the missing values of a symmetric positive semidefinite matrix
  
  The determinant of your matrix is $-(x-1)^2$. Since a positive semidefinite matrix must have nonnegative determinant, the only possible value is $x=1$. You still need to check that the matrix is ...
  • 451k
  
  ### LU factorization distribution over addition
  
  The answer is almost certainly no. The key fact to notice is that your matrix $A = I + \alpha D$ is a rank $n$ perturbation of the matrix $\alpha D$. You cannot profit from applying the Sherman-...
  • 12.7k
  Accepted
  
  ### Efficient algorithm of rank-one update of the Cholesky decomposition
  
  Tensorflow uses the implementation recorded in Krause & Igel (2015). reference: Krause, O., & Igel, C. (2015, January). A more efficient rank-one covariance matrix update for evolution ...
  Accepted
  
  ### Symmetric matrix with integer values is positive semidefinite
  
  Again this is not true. $$\det\begin{pmatrix} 9&8&7&6\\ 8&9&0&0\\ 7&0&9&0\\ 6&0&0&9 \end{pmatrix}=-5508.$$ Also, if you replace the $0$s with $1$s the ...
  • 33.3k
metadata: {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}