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Chapter 6: Problem 24

Prove that if \(A \geq 0\) then \(A\) has a unique nonnegative square root.

Short Answer

Expert verified
A nonnegative matrix \( A \) has a unique nonnegative square root due to eigenvalue decomposition and the unique square roots of nonnegative numbers.

Step by step solution

01

Understanding Nonnegative Matrices

A matrix \( A \) is said to be nonnegative, denoted as \( A \geq 0 \), if all its entries are nonnegative. We aim to show that under these circumstances, \( A \) has a unique nonnegative square root.
02

Using Spectral Theorem for Symmetric Matrices

Assume \( A \) is a nonnegative matrix. If \( A \) is symmetric, by the spectral theorem, \( A \) can be diagonalized as \( A = Q \Lambda Q^T \), where \( Q \) is an orthogonal matrix and \( \Lambda \) is a diagonal matrix with nonnegative eigenvalues, since \( A \geq 0 \).
03

Taking Square Roots of Eigenvalues

Consider \( \Lambda \) with eigenvalues \( \lambda_1, \lambda_2, \ldots, \lambda_n \). Since \( \lambda_i \geq 0 \), each eigenvalue has a unique nonnegative square root \( \sqrt{\lambda_i} \). Form a diagonal matrix \( \Lambda^{1/2} \) with entries \( \sqrt{\lambda_1}, \sqrt{\lambda_2}, \ldots, \sqrt{\lambda_n} \).
04

Constructing the Nonnegative Square Root

Using the diagonal matrix of square roots \( \Lambda^{1/2} \), construct \( A^{1/2} = Q \Lambda^{1/2} Q^T \). This matrix is symmetric and has nonnegative entries due to the construction involving nonnegative square roots.
05

Uniqueness of the Nonnegative Square Root

If there were another nonnegative matrix \( B \) such that \( B^2 = A \), then by the uniqueness of square roots for nonnegative numbers (eigenvalues), \( B \) must have the same eigenvalues as \( A^{1/2} \) and thus must be equal to \( A^{1/2} \). Hence, the nonnegative square root is unique.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spectral Theorem
The spectral theorem is a powerful tool in linear algebra, particularly when dealing with symmetric matrices. For a symmetric matrix like nonnegative matrix \( A \), the spectral theorem allows us to extract valuable information. It states that any symmetric matrix \( A \) can be expressed as \( A = Q \Lambda Q^T \), where \( Q \) is an orthogonal matrix. An orthogonal matrix is one whose inverse is its transpose, making computations easier. Additionally, \( \Lambda \) is a diagonal matrix containing the eigenvalues of \( A \).
  • The decomposition involves transforming \( A \) into a diagonal form through rotation, captured by \( Q \), keeping the underlying properties intact.
  • In the context of proving the existence of a nonnegative square root, the spectral theorem simplifies the problem to diagonal matrices, which are easier to handle.
Hence, understanding the spectral theorem is crucial for diagonalizing symmetric matrices and accessing their properties through their eigenvalues.
Eigenvalues
Eigenvalues are a fundamental concept in understanding matrices, particularly in the context of diagonalization. For a matrix \( A \), eigenvalues are special numbers associated with the matrix that provide insight into its behavior.
  • When we refer to the eigenvalues of a nonnegative symmetric matrix \( A \), we emphasize that all eigenvalues are nonnegative due to the property \( A \geq 0 \).
  • These eigenvalues are found in the diagonal matrix \( \Lambda \), resulting from the spectral theorem's representation \( A = Q \Lambda Q^T \). Each eigenvalue corresponds to a transformation along its eigenvector.
Knowing the eigenvalues not only helps in deducing the matrix's nonnegative square roots but also provides overall context to its mathematical structure, specifically through the simplicity provided by diagonalization.
Nonnegative Square Root
The nonnegative square root of a matrix involves finding a matrix \( A^{1/2} \) such that \( A^{1/2} \times A^{1/2} = A \). In the context of a nonnegative symmetric matrix \( A \), this task becomes manageable by focusing on its eigenvalues.
  • Once the matrix is diagonalized to form \( A = Q \Lambda Q^T \), we examine the nonnegative eigenvalues in \( \Lambda \).
  • Each eigenvalue \( \lambda_i \) is nonnegative, allowing a unique nonnegative square root \( \sqrt{\lambda_i} \) to exist in the diagonal matrix \( \Lambda^{1/2} \).
The nonnegative square root of the original matrix \( A \) is constructed as \( A^{1/2} = Q \Lambda^{1/2} Q^T \), ensuring the entries remain nonnegative and symmetric while proving the existence and uniqueness of such a root.
Symmetric Matrix
A symmetric matrix is one of the most widely studied types in linear algebra due to its properties and applications. For a matrix \( A \) to be symmetric, it must be equal to its transpose, which means \( A = A^T \). Such matrices have unique characteristics:
  • Symmetric matrices are guaranteed to have real eigenvalues, making them suitable candidates for spectral decomposition.
  • They make diagonalization effective through orthogonal matrices \( Q \), allowing reversible transformations without changing the eigenvalue magnitudes.
  • In the problem of finding a nonnegative square root, being symmetric plays a crucial role. It ensures the simplicity of eigenvalue calculations and the faithful preservation of matrix structures under transformations.
The properties of symmetric matrices significantly aid in the mathematical ease of analysis and computational efficiency when dealing with nonnegative matrices and their roots.

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Most popular questions from this chapter

If \(N_{1}, \ldots, N_{k}\) is a finite number of commuting normal transtormations, prove that there exists a unitary transformation \(T\) such that all of \(T N_{1} T^{-1}\) are diagonal.

If \(F\) is the field of complex numbers, evaluate the following determinants: (a) \(\left|\begin{array}{cc}1 & i \\ 2-i & 3\end{array}\right|\). (b) \(\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right|\) (c) \(\left|\begin{array}{rrrr}5 & 6 & 8 & -1 \\ 4 & 3 & 0 & 0 \\ 10 & 12 & 16 & -2 \\ 1 & 2 & 3 & 4\end{array}\right|\).

Determine the rank and signature of the following real quadratic forms: (a) \(x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}\). (b) \(x_{1}{ }^{2}+x_{1} x_{2}+2 x_{1} x_{3}+2 x_{2}{ }^{2}+4 x_{2} x_{3}+2 x_{3}{ }^{2}\).

If \(F\) is the real field, prove that it is impossible to find matrices \(A, B \in F_{n}\) such that \(A B-B A=1\).

If \(F\) is a field of characteristic different from 2, given \(A \in F_{n}\), prove that there exists a \(B \in F_{n}\) such that \(B A B^{\prime}\) is diagonal.
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