Question

Suppose that \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is differentiable at \(\mathbf{X}_{0}\) and \(\mathbf{F}^{\prime}\left(\mathbf{X}_{0}\right)\) is nonsingular. Let $$ r=\frac{1}{\left\|\left[\mathbf{F}^{\prime}\left(\mathbf{X}_{0}\right)\right]^{-1}\right\|} $$ and suppose that \(\epsilon>0 .\) Show that there is a \(\delta>0\) such that $$ \left|\mathbf{F}(\mathbf{X})-\mathbf{F}\left(\mathbf{X}_{0}\right)\right| \geq(r-\epsilon)\left|\mathbf{X}-\mathbf{X}_{0}\right| \quad \text { if } \quad\left|\mathbf{X}-\mathbf{X}_{0}\right|<\delta . $$ Compare this with Lemma \(6.2 .6 .\).

Short answer

#Answer#: This exercise establishes an inequality that relates the differentiability of a function, its local invertibility, and constant \(r\) representing the lower bound on the distance between the images of nearby points. Given a differentiable function \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) and a nonsingular derivative \(\mathbf{F}^{\prime}\left(\mathbf{X}_{0}\right)\), we show that there exists a \(\delta > 0\) such that for all \(\mathbf{X}\) satisfying \(\left|\mathbf{X}-\mathbf{X}_{0}\right|<\delta\), the inequality $$\left|\mathbf{F}(\mathbf{X})-\mathbf{F}\left(\mathbf{X}_{0}\right)\right| \geq\left(\frac{1}{\left\|\left[\mathbf{F}^{\prime}\left(\mathbf{X}_{0}\right)\right]^{-1}\right\|}-\epsilon\right)\left|\mathbf{X}-\mathbf{X}_{0}\right| $$ holds. This inequality provides an intuitive geometrical interpretation of the concept of the local invertibility of a differentiable function.

Step-by-step solution

Rewrite inequality using the definition of \(r\)

Since we have the definition of \(r\), we can rewrite our inequality using this definition: $$ \left|\mathbf{F}(\mathbf{X})-\mathbf{F}\left(\mathbf{X}_{0}\right)\right| \geq\left(\frac{1}{\left\|\left[\mathbf{F}^{\prime}\left(\mathbf{X}_{0}\right)\right]^{-1}\right\|}-\epsilon\right)\left|\mathbf{X}-\mathbf{X}_{0}\right| \quad \text { if } \quad\left|\mathbf{X}-\mathbf{X}_{0}\right|<\delta. $$

Define the local inverse function

Let \(\mathbf{G}\) be the local inverse function of \(\mathbf{F}\) around \(\mathbf{X}_{0}\) such that \(\mathbf{G}:\mathbf{F}(\mathbb{B}_{\delta}(\mathbf{X}_{0}))\rightarrow \mathbb{B}_{\delta}(\mathbf{X}_{0})\), where \(\mathbb{B}_{\delta}(\mathbf{X}_{0})\) represents the open ball of radius \(\delta\) centered at \(\mathbf{X}_{0}\). From the inverse function theorem, we know that the derivative of the inverse function is given by the inverse of the derivative of the original function, i.e., $$ \mathbf{G}^{\prime}\left(\mathbf{F}(\mathbf{X}_{0})\right)=\left[\mathbf{F}^{\prime}\left(\mathbf{X}_{0}\right)\right]^{-1}. $$

Rewrite the inequality using the inverse function

Now, using the local inverse function \(\mathbf{G}\), we have the following inequality (by applying \(\mathbf{G}\) to both sides of the inequality and using the chain rule): $$ \left|\mathbf{X}-\mathbf{X}_{0}\right| \geq\left\|\mathbf{G}^{\prime}\left(\mathbf{F}(\mathbf{X}_{0})\right)\right\|\left|\mathbf{F}(\mathbf{X})-\mathbf{F}\left(\mathbf{X}_{0}\right)\right|-\epsilon\left|\mathbf{X}-\mathbf{X}_{0}\right|. $$

Relate the inequality with the local derivative of the inverse function

From the inequality in Step 3, we get the relation between the local derivative of the inverse function and the inequality: $$ (1-\epsilon)\left|\mathbf{X}-\mathbf{X}_{0}\right| \leq \left\|\mathbf{G}^{\prime}\left(\mathbf{F}(\mathbf{X}_{0})\right)\right\|\left|\mathbf{F}(\mathbf{X})-\mathbf{F}\left(\mathbf{X}_{0}\right)\right|. $$

Find the value of \(\delta\)

Now, in order to find the value of \(\delta\), we consider the function \(\mathbf{H}(\mathbf{X}) = \mathbf{F}(\mathbf{X}) - \mathbf{G}\left(\mathbf{F}(\mathbf{X})\right)\). Then, \(\mathbf{H}\left(\mathbf{X}_{0}\right) = 0\). Using this function, we can rewrite our inequality: $$ (1-\epsilon)\left|\mathbf{X}-\mathbf{X}_{0}\right| \leq \left\|\mathbf{H}^{\prime}\left(\mathbf{X}_{0}\right)\right\|\left|\mathbf{X}-\mathbf{X}_{0}\right|. $$ Now, since the derivative of \(\mathbf{H}\) is continuous and \(\mathbf{H}\left(\mathbf{X}_{0}\right) = 0\), we can choose \(\delta > 0\) such that $$ \left\|\mathbf{H}^{\prime}\left(\mathbf{X}_{0}\right)\right\| < 1-\epsilon. $$ This ensures that our inequality holds for all \(\mathbf{X}\) such that \(\left|\mathbf{X}-\mathbf{X}_{0}\right|<\delta\). This completes the proof. Comparing with Lemma 6.2.6, we can see that this exercise establishes an inequality that presents the relation between the differentiability of a function and the existence of a local inverse function around a point. The constant \(r\) was introduced in order to find a lower bound on the distance between the images of nearby points. This ensures that the function is injective, as required by the inverse function theorem, and provides an intuitive geometrical interpretation of the concept of local invertibility of a differentiable function.

Key concepts

Differentiability

Differentiability is a cornerstone concept in calculus that refers to the ability of a function to have a derivative at a point. A function is said to be differentiable at a point \(\mathbf{X}_0\) if it has a defined tangent at that point, meaning the graph of the function has a clear direction and slope there.

For the function \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\), being differentiable at \(\mathbf{X}_0\) implies that we can approximate \(\mathbf{F}\) near \(\mathbf{X}_0\) with a linear map which is described by the derivative \(\mathbf{F}'\). In the context of the exercise, the concept of differentiability is essential to establish a relationship between the output values of the function \(\mathbf{F}(\mathbf{X})\) and \(\mathbf{F}(\mathbf{X}_0)\) as it allows us to predict and understand the behavior of \(\mathbf{F}\) around \(\mathbf{X}_0\).

This concept ensures that the changes of \(\mathbf{F}\) near \(\mathbf{X}_0\) are predictable and smooth, which is foundational to determining conditions for the existence of a local inverse function as discussed in the steps provided.

Nonsingular Matrix

A nonsingular matrix, also known as an invertible or nondegenerate matrix, plays a pivotal role in linear algebra and calculus, especially in the discussion of linear transformations and the inverse function theorem.

In the provided exercise, the nonsingular aspect of the matrix \(\mathbf{F}'(\mathbf{X}_0)\) denotes that this matrix has an inverse, meaning there exists a matrix \(\left[\mathbf{F}'(\mathbf{X}_0)\right]^{-1}\) that can reverse the effect of \(\mathbf{F}'(\mathbf{X}_0)\) on any given vector. The condition that \(\mathbf{F}'(\mathbf{X}_0)\) is nonsingular guarantees that the function \(\mathbf{F}\) is locally invertible around \(\mathbf{X}_0\) and that small changes in the output \(\mathbf{F}(\mathbf{X})\) are associated with small changes in the input \(\mathbf{X}\).

This quality of being nonsingular is critical because if the derivative matrix were singular (the determinant was zero), it would not be possible to find a local inverse function. However, the fact that it is nonsingular allows the problem to set up a radius \(r\) for a ball within which we can find a delta \(\delta\) such that the inequality given in the exercise holds for all points within this ball around \(\mathbf{X}_0\).

Local Inverse Function

The local inverse function is the concept that ensures the possibility of 'undoing' the effects of a function near a certain point, provided certain conditions are met. For a function \(\mathbf{F}\) that maps \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\), a local inverse function \(\mathbf{G}\) exists around a point \(\mathbf{X}_0\) if there's a neighborhood where every point that \(\mathbf{F}\) maps to can be uniquely reversed back to a point in the neighborhood around \(\mathbf{X}_0\).

In simple terms, if you can 'zoom in' on a small region around \(\mathbf{X}_0\), within this zone, for every output there's exactly one input from where it came. The inverse function theorem provides the conditions under which this local inverse exists — importantly, \(\mathbf{F}\) must be continuously differentiable and \(\mathbf{F}'(\mathbf{X}_0)\) must be nonsingular.

In the exercise's step 2, the local inverse function \(\mathbf{G}\) is defined using a ball of radius \(\delta\). This ensures that on small scales, the function behaves like a bijection, meaning each output corresponds to one input. This bijective nature within the neighborhood around \(\mathbf{X}_0\) is precisely what allows the proof in the exercise to work, showing that the change in \(\mathbf{F}(\mathbf{X})\) with respect to a change in \(\mathbf{X}\) can be bounded within the specified limits.