# Complex matrix without a square root

Consider for $$n \ge 2$$ the linear space $$\mathcal M_n(\mathbb C)$$ of complex matrices of dimension $$n \times n$$. Is a matrix $$T \in \mathcal M_n(\mathbb C)$$ always having a square root $$S \in \mathcal M_n(\mathbb C)$$, i.e. a matrix such that $$S^2=T$$? is the question we deal with.

First, one can note that if $$T$$ is similar to $$V$$ with $$T = P^{-1} V P$$ and $$V$$ has a square root $$U$$ then $$T$$ also has a square root as $$V=U^2$$ implies $$T=\left(P^{-1} U P\right)^2$$.

### Diagonalizable matrices

Suppose that $$T$$ is similar to a diagonal matrix $D=\begin{bmatrix} d_1 & 0 & \dots & 0 \\ 0 & d_2 & \dots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & \dots & d_n \end{bmatrix}$ Any complex number has two square roots, except $$0$$ which has only one. Therefore, each $$d_i$$ has at least one square root $$d_i^\prime$$ and the matrix $D^\prime=\begin{bmatrix} d_1^\prime & 0 & \dots & 0 \\ 0 & d_2^\prime & \dots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & \dots & d_n^\prime \end{bmatrix}$ is a square root of $$D$$. Continue reading Complex matrix without a square root

# Intersection and union of interiors

Consider a topological space $$E$$. For subsets $$A, B \subseteq E$$ we have the equality $A^\circ \cap B^\circ = (A \cap B)^\circ$ and the inclusion $A^\circ \cup B^\circ \subseteq (A \cup B)^\circ$ where $$A^\circ$$ and $$B^\circ$$ denote the interiors of $$A$$ and $$B$$.

Let’s prove that $$A^\circ \cap B^\circ = (A \cap B)^\circ$$.

We have $$A^\circ \subseteq A$$ and $$B^\circ \subseteq B$$ and therefore $$A^\circ \cap B^\circ \subseteq A \cap B$$. As $$A^\circ \cap B^\circ$$ is open we then have $$A^\circ \cap B^\circ \subseteq (A \cap B)^\circ$$ because $$A^\circ \cap B^\circ$$ is open and $$(A \cap B)^\circ$$ is the largest open subset of $$A \cap B$$.

Conversely, $$A \cap B \subseteq A$$ implies $$(A \cap B)^\circ \subseteq A^\circ$$ and similarly $$(A \cap B)^\circ \subseteq B^\circ$$. Therefore we have $$(A \cap B)^\circ \subseteq A^\circ \cap B^\circ$$ which concludes the proof of the equality $$A^\circ \cap B^\circ = (A \cap B)^\circ$$.

One can also prove the inclusion $$A^\circ \cup B^\circ \subseteq (A \cup B)^\circ$$. However, the equality $$A^\circ \cup B^\circ = (A \cup B)^\circ$$ doesn’t always hold. Let’s provide a couple of counterexamples.

For the first one, let’s take for $$E$$ the plane $$\mathbb R^2$$ endowed with usual topology. For $$A$$, we take the unit close disk and for $$B$$ the plane minus the open unit disk. $$A^\circ$$ is the unit open disk and $$B^\circ$$ the plane minus the unit closed disk. Therefore $$A^\circ \cup B^\circ = \mathbb R^2 \setminus C$$ is equal to the plane minus the unit circle $$C$$. While we have $A \cup B = (A \cup B)^\circ = \mathbb R^2.$

For our second counterexample, we take $$E=\mathbb R$$ endowed with usual topology and $$A = \mathbb R \setminus \mathbb Q$$, $$B = \mathbb Q$$. Here we have $$A^\circ = B^\circ = \emptyset$$ thus $$A^\circ \cup B^\circ = \emptyset$$ while $$A \cup B = (A \cup B)^\circ = \mathbb R$$.

The union of the interiors of two subsets is not always equal to the interior of the union.

# Additive subgroups of vector spaces

Consider a vector space $$V$$ over a field $$F$$. A subspace $$W \subseteq V$$ is an additive subgroup of $$(V,+)$$. The converse might not be true.

If the characteristic of the field is zero, then a subgroup $$W$$ of $$V$$ might not be an additive subgroup. For example $$\mathbb R$$ is a vector space over $$\mathbb R$$ itself. $$\mathbb Q$$ is an additive subgroup of $$\mathbb R$$. However $$\sqrt{2}= \sqrt{2}.1 \notin \mathbb Q$$ proving that $$\mathbb Q$$ is not a subspace of $$\mathbb R$$.

Another example is $$\mathbb Q$$ which is a vector space over itself. $$\mathbb Z$$ is an additive subgroup of $$\mathbb Q$$, which is not a subspace as $$\frac{1}{2} \notin \mathbb Z$$.

Yet, an additive subgroup of a vector space over a prime field $$\mathbb F_p$$ with $$p$$ prime is a subspace. To prove it, consider an additive subgroup $$W$$ of $$(V,+)$$ and $$x \in W$$. For $$\lambda \in F$$, we can write $$\lambda = \underbrace{1 + \dots + 1}_{\lambda \text{ times}}$$. Consequently $\lambda \cdot x = (1 + \dots + 1) \cdot x= \underbrace{x + \dots + x}_{\lambda \text{ times}} \in W.$

Finally an additive subgroup of a vector space over any finite field is not always a subspace. For a counterexample, take the non-prime finite field $$\mathbb F_{p^2}$$ (also named $$\text{GF}(p^2)$$). $$\mathbb F_{p^2}$$ is also a vector space over itself. The prime finite field $$\mathbb F_p \subset \mathbb F_{p^2}$$ is an additive subgroup that is not a subspace of $$\mathbb F_{p^2}$$.

# A differentiable real function with unbounded derivative around zero

Consider the real function defined on $$\mathbb R$$$f(x)=\begin{cases} 0 &\text{for } x = 0\\ x^2 \sin \frac{1}{x^2} &\text{for } x \neq 0 \end{cases}$

$$f$$ is continuous and differentiable on $$\mathbb R\setminus \{0\}$$. For $$x \in \mathbb R$$ we have $$\vert f(x) \vert \le x^2$$, which implies that $$f$$ is continuous at $$0$$. Also $\left\vert \frac{f(x)-f(0)}{x} \right\vert = \left\vert x \sin \frac{1}{x^2} \right\vert \le \vert x \vert$ proving that $$f$$ is differentiable at zero with $$f^\prime(0) = 0$$. The derivative of $$f$$ for $$x \neq 0$$ is $f^\prime(x) = \underbrace{2x \sin \frac{1}{x^2}}_{=g(x)}-\underbrace{\frac{2}{x} \cos \frac{1}{x^2}}_{=h(x)}$ On the interval $$(-1,1)$$, $$g(x)$$ is bounded by $$2$$. However, for $$a_k=\frac{1}{\sqrt{k \pi}}$$ with $$k \in \mathbb N$$ we have $$h(a_k)=2 \sqrt{k \pi} (-1)^k$$ which is unbounded while $$\lim\limits_{k \to \infty} a_k = 0$$. Therefore $$f^\prime$$ is unbounded in all neighborhood of the origin.

# A Riemann-integrable map that is not regulated

For a Banach space $$X$$, a function $$f : [a,b] \to X$$ is said to be regulated if there exists a sequence of step functions $$\varphi_n : [a,b] \to X$$ converging uniformly to $$f$$.

One can prove that a regulated function $$f : [a,b] \to X$$ is Riemann-integrable. Is the converse true? The answer is negative and we provide below an example of a Riemann-integrable real function that is not regulated. Let’s first prove following theorem.

THEOREM A bounded function $$f : [a,b] \to \mathbb R$$ that is (Riemann) integrable on all intervals $$[c, b]$$ with $$a < c < b$$ is integrable on $$[a,b]$$.

PROOF Take $$M > 0$$ such that for all $$x \in [a,b]$$ we have $$\vert f(x) \vert < M$$. For $$\epsilon > 0$$, denote $$c = \inf(a + \frac{\epsilon}{4M},b + \frac{b-a}{2})$$. As $$f$$ is supposed to be integrable on $$[c,b]$$, one can find a partition $$P$$: $$c=x_1 < x_2 < \dots < x_n =b$$ such that $$0 \le U(f,P) - L(f,P) < \frac{\epsilon}{2}$$ where $$L(f,P),U(f,P)$$ are the lower and upper Darboux sums. For the partition $$P^\prime$$: $$a= x_0 < c=x_1 < x_2 < \dots < x_n =b$$, we have \begin{aligned} 0 \le U(f,P^\prime) - L(f,P^\prime) &\le 2M(c-a) + \left(U(f,P) - L(f,P)\right)\\ &< 2M \frac{\epsilon}{4M} + \frac{\epsilon}{2} = \epsilon \end{aligned} We now prove that the function $$f : [0,1] \to [0,1]$$ defined by $f(x)=\begin{cases} 1 &\text{ if } x \in \{2^{-k} \ ; \ k \in \mathbb N\}\\ 0 &\text{otherwise} \end{cases}$ is Riemann-integrable (that follows from above theorem) and not regulated. Let's prove it. If $$f$$ was regulated, there would exist a step function $$g$$ such that $$\vert f(x)-g(x) \vert < \frac{1}{3}$$ for all $$x \in [0,1]$$. If $$0=x_0 < x_1 < \dots < x_n=1$$ is a partition associated to $$g$$ and $$c_1$$ the value of $$g$$ on the interval $$(0,x_1)$$, we must have $$\vert 1-c_1 \vert < \frac{1}{3}$$ as $$f$$ takes (an infinite number of times) the value $$1$$ on $$(0,x_1)$$. But $$f$$ also takes (an infinite number of times) the value $$0$$ on $$(0,x_1)$$. Hence we must have $$\vert c_1 \vert < \frac{1}{3}$$. We get a contradiction as those two inequalities are not compatible.

# A discontinuous midpoint convex function

Let’s recall that a real function $$f: \mathbb R \to \mathbb R$$ is called convex if for all $$x, y \in \mathbb R$$ and $$\lambda \in [0,1]$$ we have $f((1- \lambda) x + \lambda y) \le (1- \lambda) f(x) + \lambda f(y)$ $$f$$ is called midpoint convex if for all $$x, y \in \mathbb R$$ $f \left(\frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2}$ One can prove that a continuous midpoint convex function is convex. Sierpinski proved the stronger theorem, that a real-valued Lebesgue measurable function that is midpoint convex will be convex.

Can one find a discontinuous midpoint convex function? The answer is positive but requires the Axiom of Choice. Why? Because Robert M. Solovay constructed a model of Zermelo-Fraenkel set theory (ZF), exclusive of the axiom of choice where all functions are Lebesgue measurable. Hence convex according to Sierpinski theorem. And one knows that convex functions defined on $$\mathbb R$$ are continuous.

Referring to my previous article on the existence of discontinuous additive map, let’s use a Hamel basis $$\mathcal B = (b_i)_{i \in I}$$ of $$\mathbb R$$ considered as a vector space on $$\mathbb Q$$. Take $$i_1 \in I$$, define $$f(i_1)=1$$ and $$f(i)=0$$ for $$i \in I\setminus \{i_1\}$$ and extend $$f$$ linearly on $$\mathbb R$$. $$f$$ is midpoint convex as it is linear. As the image of $$\mathbb R$$ under $$f$$ is $$\mathbb Q$$, $$f$$ is discontinuous as explained in the discontinuous additive map counterexample.

Moreover, $$f$$ is unbounded on all open real subsets. By linearity, it is sufficient to prove that $$f$$ is unbounded around $$0$$. Let’s consider $$i_1 \neq i_2 \in I$$. $$G= b_{i_1} \mathbb Z + b_{i_2} \mathbb Z$$ is a proper subgroup of the additive $$\mathbb R$$ group. Hence $$G$$ is either dense of discrete. It cannot be discrete as the set of vectors $$\{b_1,b_2\}$$ is linearly independent. Hence $$G$$ is dense in $$\mathbb R$$. Therefore, one can find a non vanishing sequence $$(x_n)_{n \in \mathbb N}=(q_n^1 b_{i_1} + q_n^2 b_{i_2})_{n \in \mathbb N}$$ (with $$(q_n^1,q_n^2) \in \mathbb Q^2$$ for all $$n \in \mathbb N$$) converging to $$0$$. As $$\{b_1,b_2\}$$ is linearly independent, this implies $$\vert q_n^1 \vert, \vert q_n^2 \vert \underset{n\to+\infty}{\longrightarrow} \infty$$ and therefore $\lim\limits_{n \to \infty} \vert f(x_n) \vert = \lim\limits_{n \to \infty} \vert f(q_n^1 b_{i_1} + q_n^2 b_{i_2}) \vert = \lim\limits_{n \to \infty} \vert q_n^1 \vert = \infty.$

# A discontinuous additive map

A function $$f$$ defined on $$\mathbb R$$ into $$\mathbb R$$ is said to be additive if and only if for all $$x, y \in \mathbb R$$
$f(x+y) = f(x) + f(y).$ If $$f$$ is supposed to be continuous at zero, $$f$$ must have the form $$f(x)=cx$$ where $$c=f(1)$$. This can be shown using following steps:

• $$f(0) = 0$$ as $$f(0) = f(0+0)= f(0)+f(0)$$.
• For $$q \in \mathbb N$$ $$f(1)=f(q \cdot \frac{1}{q})=q f(\frac{1}{q})$$. Hence $$f(\frac{1}{q}) = \frac{f(1)}{q}$$. Then for $$p,q \in \mathbb N$$, $$f(\frac{p}{q}) = p f(\frac{1}{q})= f(1) \frac{p}{q}$$.
• As $$f(-x) = -f(x)$$ for all $$x \in\mathbb R$$, we get that for all rational number $$\frac{p}{q} \in \mathbb Q$$, $$f(\frac{p}{q})=f(1)\frac{p}{q}$$.
• The equality $$f(x+y) = f(x) + f(y)$$ implies that $$f$$ is continuous on $$\mathbb R$$ if it is continuous at $$0$$.
• We can finally conclude to $$f(x)=cx$$ for all real $$x \in \mathbb R$$ as the rational numbers are dense in $$\mathbb R$$.

We’ll use a Hamel basis to construct a discontinuous linear function. The set $$\mathbb R$$ can be endowed with a vector space structure over $$\mathbb Q$$ using the standard addition and the multiplication by a rational for the scalar multiplication.

Using the axiom of choice, one can find a (Hamel) basis $$\mathcal B = (b_i)_{i \in I}$$ of $$\mathbb R$$ over $$\mathbb Q$$. That means that every real number $$x$$ is a unique linear combination of elements of $$\mathcal B$$: $x= q_1 b_{i_1} + \dots + q_n b_{i_n}$ with rational coefficients $$q_1, \dots, q_n$$. The function $$f$$ is then defined as $f(x) = q_1 + \dots + q_n.$ The linearity of $$f$$ follows from its definition. $$f$$ is not continuous as it only takes rational values which are not all equal. And one knows that the image of $$\mathbb R$$ under a continuous map is an interval.

# Existence of a continuous function with divergent Fourier series

In that article, I provided an example of a continuous function with divergent Fourier series. We prove here the existence of such a function using Banach-Steinhaus theorem, also called uniform boundedness principle.

Theorem (Uniform Boundedness Theorem) Let $$(X, \Vert \cdot \Vert_X)$$ be a Banach space and $$(Y, \Vert \cdot \Vert_Y)$$ be a normed vector space. Suppose that $$F$$ is a set of continuous linear operators from $$X$$ to $$Y$$. If for all $$x \in X$$ one has $\sup\limits_{T \in F} \Vert T(x) \Vert_Y \lt \infty$ then $\sup\limits_{T \in F, \ \Vert x \Vert = 1} \Vert T(x) \Vert_Y \lt \infty$

Let’s take for $$X$$ the vector space $$\mathcal C_{2 \pi}$$ of continuous functions from $$\mathbb R$$ to $$\mathbb C$$ which are periodic with period $$2 \pi$$ endowed with the norm $$\Vert f \Vert_\infty = \sup\limits_{- \pi \le t \le \pi} \vert f(t) \vert$$. $$(\mathcal C_{2 \pi}, \Vert \cdot \Vert_\infty)$$ is a Banach space. For the vector space $$Y$$, we take the complex numbers $$\mathbb C$$ endowed with the modulus.

For $$n \in \mathbb N$$, the map $\begin{array}{l|rcl} \ell_n : & \mathcal C_{2 \pi} & \longrightarrow & \mathbb C \\ & f & \longmapsto & \displaystyle \sum_{p=-n}^n c_p(f) \end{array}$ is a linear operator, where for $$p \in \mathbb Z$$, $$c_p(f)$$ denotes the complex Fourier coefficient $c_p(f) = \frac{1}{2 \pi} \int_{- \pi}^{\pi} f(t) e^{-i p t} \ dt$

We now prove that
\begin{align*}
\Lambda_n &= \sup\limits_{f \in \mathcal C_{2 \pi}, \Vert f \Vert_\infty=1} \vert \ell_n(f) \vert\\
&= \frac{1}{2 \pi} \int_{- \pi}^{\pi} \left\vert \frac{\sin (2n+1)\frac{t}{2}}{\sin \frac{t}{2}} \right\vert \ dt = \frac{1}{2 \pi} \int_{- \pi}^{\pi} \left\vert h_n(t) \right\vert \ dt,
\end{align*} where one can notice that the function $\begin{array}{l|rcll} h_n : & [- \pi, \pi] & \longrightarrow & \mathbb C \\ & t & \longmapsto & \frac{\sin (2n+1)\frac{t}{2}}{\sin \frac{t}{2}} &\text{for } t \neq 0\\ & 0 & \longmapsto & 2n+1 \end{array}$ is continuous.
Continue reading Existence of a continuous function with divergent Fourier series

# A positive smooth function with all derivatives vanishing at zero

Let’s consider the set $$\mathcal C^\infty(\mathbb R)$$ of real smooth functions, i.e. functions that have derivatives of all orders on $$\mathbb R$$.

Does a positive function $$f \in \mathcal C^\infty(\mathbb R)$$ with all derivatives vanishing at zero exists?

Such a map $$f$$ cannot be expandable in power series around zero, as it would vanish in a neighborhood of zero. However, the answer to our question is positive and we’ll prove that $f(x) = \left\{\begin{array}{lll} e^{-\frac{1}{x^2}} &\text{if} &x \neq 0\\ 0 &\text{if} &x = 0 \end{array}\right.$ provides an example.

$$f$$ is well defined and positive for $$x \neq 0$$. As $$\lim\limits_{x \to 0} -\frac{1}{x^2} = -\infty$$, we get $$\lim\limits_{x \to 0} f(x) = 0$$ proving that $$f$$ is continuous on $$\mathbb R$$. Let’s prove by induction that for $$x \neq 0$$ and $$n \in \mathbb N$$, $$f^{(n)}(x)$$ can be written as $f^{(n)}(x) = \frac{P_n(x)}{x^{3n}}e^{-\frac{1}{x^2}}$ where $$P_n$$ is a polynomial function. The statement is satisfied for $$n = 1$$ as $$f^\prime(x) = \frac{2}{x^3}e^{-\frac{1}{x^2}}$$. Suppose that the statement is true for $$n$$ then $f^{(n+1)}(x)=\left[\frac{P_n^\prime(x)}{x^{3n}} – \frac{3n P_n(x)}{x^{3n+1}}+\frac{2 P_n(x)}{x^{3n+3}}\right] e^{-\frac{1}{x^2}}$ hence the statement is also true for $$n+1$$ by taking $$P_{n+1}(x)= x^3 P_n^\prime(x) – 3n x^2 P_n(x) + 2 P_n(x)$$. Which concludes our induction proof.

Finally, we have to prove that for all $$n \in \mathbb N$$, $$\lim\limits_{x \to 0} f^{(n)}(x) = 0$$. For that, we use the power expansion of the exponential map $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$. For $$x \neq 0$$, we have $\left\vert x \right\vert^{3n} e^{\frac{1}{x^2}} \ge \frac{\vert x \vert^{3n}}{(2n)! \vert x \vert ^{4n}} = \frac{1}{(2n)! \vert x \vert^n}$ Therefore $$\lim\limits_{x \to 0} \left\vert x \right\vert^{3n} e^{\frac{1}{x^2}} = \infty$$ and $$\lim\limits_{x \to 0} f^{(n)}(x) = 0$$ as $$f^{(n)}(x) = \frac{P_n(x)}{x^{3n} e^{\frac{1}{x^2}}}$$ with $$P_n$$ a polynomial function.

# Non commutative rings

Let’s recall that a set $$R$$ equipped with two operations $$(R,+,\cdot)$$ is a ring if and only if $$(R,+)$$ is an abelian group, multiplication $$\cdot$$ is associative and has a multiplicative identity $$1$$ and multiplication is left and right distributive with respect to addition.

$$(\mathbb Z, +, \cdot)$$ is a well known infinite ring which is commutative. The rational, real and complex numbers are other infinite commutative rings. Those are in fact fields as every non-zero element have a multiplicative inverse.

For a field $$F$$ (finite or infinite), the polynomial ring $$F[X]$$ is another example of infinite commutative ring.

Also for $$n$$ integer, the integers modulo n is a finite ring that is commutative. Finally, according to Wedderburn theorem every finite division ring is commutative.

So what are examples of non commutative rings? Let’s provide a couple. Continue reading Non commutative rings