Intereting Posts

I found this odd relationship, $x^2 = \sum_\limits{k = 0}^{x-1} (2k + 1)$.
Global sections on quasi coherent sheaves on affine scheme
Constructing a set with exactly three limit points
Proving irrationals are dense in (0, 1)
Intuitive or visual understanding of the real projective plane
Two problems about Squarefree numbers
$3^2 \ 5^2 \ldots (p-2)^2 \equiv (-1)^{\frac{p+1}{2}} \ (\mathrm{mod} \ p)$
If capital letters are supposed to be sets, why is $N$ used as a number?
Why is the Riemann sum less than the value of the integral?
Bounded operators that are not closed.
Covariance inequality for $n$ exchangeable random variables
Quick question on abundant numbers
Is the square root of a triangular matrix necessarily triangular?
Composition of a piecewise and non-piecewise function
Fourier transform of even/odd function

Let $A$ $\in$ $ \mathbb{R} ^{n×n}$ and consider the series:

$$S = \sum_{k=0}^{\infty} A^{k}$$

Prove that the series converges iff all the eigenvalues of $A$ are strictly smaller than 1. Further, if the series converges, show that

$S$ is invertible with its inverse being $I − A$.

- Do non-square matrices have eigenvalues?
- What is the relation between rank of a matrix, its eigenvalues and eigenvectors
- How do I calculate generalized eigenvectors?
- A simple explanation of eigenvectors and eigenvalues with 'big picture' ideas of why on earth they matter
- Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$
- SVD and the columns — I did this wrong but it seems that it still works, why?

- linear least squares minimizing distance from points to rays - is it possible?
- Proving that $\det(A) = 0$ when the columns are linearly dependent
- Finding Boolean/Logical Expressions for truth tables in algebraic normal form(ANF)
- Block inverse of symmetric matrices
- What exactly is a basis in linear algebra?
- orthogonal group of a quadratic vector space
- Finding a unit vector perpendicular to another vector
- Determinant of a specific circulant matrix, $A_n$
- Non-Symmetric Positive Definite Matrices
- Subspaces and annihilators

$$S=I+A+A^2+A^3+A^4+…+A^n\\ \to (I-A)S=\\I(I+A+A^2+A^3+A^4+…+A^n)-A(I+A+A^2+A^3+A^4+…+A^n)=\\I+A+A^2+A^3+A^4+…+A^n-(A+A^2+A^3+A^4+…+A^n+A^{n+1})=\\I-A^{n+1}$$

so now $n \to \infty ,|\lambda_i|<1 \to $ we have $ A^{n+1} \to 0$

$$(I-A)S=I-A^{n+1} \to I\\ (I-A)S=I \to S^{-1}=(I-A)\\$$

- When does the integral preserve strict inequalities?
- Why is Lebesgue so often spelled “Lebesque”?
- When does the $f^{(n)}$ converge to a limit function as $n\to\infty$?
- Solve trigonometric equation: $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$
- Closure of a connected set is connected
- The asymptotic behavior of the CDF of Binomial distribution
- Probability that two randomly chosen permutations will generate $S_n$.
- Upper semi continuous, lower semi continuous
- Linear Algebra determinant reduction
- Formula for trace of compact operators on $L^2(\mathbb{R})$ given by integral kernels?
- The big $O$ versus little $o$ notation.
- How to find an end point of an arc given another end point, radius, and arc direction?
- What is the meaning of expressions of the type $f(\cdot)$ (function (dot))?
- Why do we use both sets and predicates?
- Why limits work