# 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
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