Algebra

Complex matrix without a square root

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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 →

Topology

Intersection and union of interiors

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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.

Algebra

Additive subgroups of vector spaces

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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 \(\mathbb F_{p^2}\).

Analysis

A differentiable real function with unbounded derivative around zero

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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.

Analysis

A Riemann-integrable map that is not regulated

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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.

Analysis

A discontinuous midpoint convex function

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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 \[
\lim\limits_{n \to \infty} \vert f(x_n) \vert = \lim\limits_{n \to \infty} \vert f(q_n^1 b_{i_1} + q_n^2 b_{i_2}) \vert = \lim\limits_{n \to \infty} \vert q_n^1 \vert = \infty.\]

Analysis

A discontinuous additive map

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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.

Analysis

Existence of a continuous function with divergent Fourier series

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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 →

Analysis

A positive smooth function with all derivatives vanishing at zero

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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.

Algebra

Non commutative rings

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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 →

Mathematical exceptions to the rules or intuition