How to prove that $X^{T}X$ is singular if $X$ is singular.
89729
If $X$ is a singular matrix, then is $X'X$ singular?
71437
How is the derivative of $\textrm{Trace}\left\{ X^T A X B\right\}$ with respect to $X$ equal to $AXB + A^TXB^T$?
14161
Proof of when is $A=X^TX$ invertible?
73677
If the matrix $A^3 = 0$, is $A$ singular?
67058
Prove that $\text{rank}(X^TX)=\text{rank}(X)$
65395
Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
68724
$||T-I|| < 1$ implies that $T$ is invertible.
9781
How to prove that $\lim_{n \to \infty} n x^{n} = 0 $ when $0<x<1$?
136195
Stuck trying to prove that $e^{-x^{-2}}$ is $C^{\infty}$
133266
If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis
11715
Prove that if $\langle Tx,x\rangle =0$ for all $x \in X$, then $T = 0$
128457
Show that $T$ is a compact operator
52614
(exercise from Tao's analysis book) Proof of a lemma relating to power set of X
155974
Show $||Tx||_2 \leq ||(v_n)||_2 \cdot ||x||_{\infty}$
49438
Prove that $X^\ast$ separable implies $X$ separable
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
36567
$A^{T}A$ positive definite then A is invertible?