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Magazine Chapter 5: Problem 54
Let \(A\) be a positive definite symmetric matrix. Show that there is a positive definite symmetric matrix \(B\) such that \(A=B^{2} .\) (Such a matrix \(B\) is called a square root of \(A .)\)
Short Answer
Expert verified
There exists a positive definite symmetric matrix \(B\) such that \(A = B^2\).
Step by step solution
01
Understanding the Problem
We need to show that for a given positive definite symmetric matrix \(A\), there exists another positive definite symmetric matrix \(B\) such that \(A = B^2\). This means \(B\) is the square root of \(A\).
02
Diagonalization of Matrix A
Since \(A\) is positive definite and symmetric, it can be diagonalized. This means there exists an orthogonal matrix \(Q\) and a diagonal matrix \(D\) with positive entries such that \(A = QDQ^T\).
03
Finding the Square Root of the Diagonal Matrix D
For the diagonal matrix \(D\), consider \(D^{1/2}\), which is a diagonal matrix where each entry is the square root of the corresponding entry in \(D\). Since all entries of \(D\) are positive, the square roots are well-defined and positive.
04
Construct the Matrix B
Define the matrix \(B\) as \(B = QD^{1/2}Q^T\). Here, \(Q\) is orthogonal, and \(D^{1/2}\) is diagonal with positive entries. Since \(A = QDQ^T\), when we square \(B\), we effectively revert back to \(A\).
05
Verify that B is Positive Definite and Symmetric
A matrix \(B = QD^{1/2}Q^T\) has the form of a symmetric matrix, and it is positive definite because the eigenvalues (entries of \(D^{1/2}\)) are positive. As \(Q\) is orthogonal, \(B\) remains symmetric and positive definite.
06
Check the Condition A = B^2
To verify \(B^2 = A\), compute \(B^2 = (QD^{1/2}Q^T)(QD^{1/2}Q^T) = QD^{1/2}(Q^TQ)D^{1/2}Q^T = QD^{1/2}D^{1/2}Q^T = QDQ^T = A\). Thus, \(B^2 = A\), confirming \(B\) is the square root of \(A\).
07
Conclusion
There exists a positive definite symmetric matrix \(B\) such that \(A = B^2\). Hence, \(B\) is indeed the square root of \(A\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Symmetric Matrix
A symmetric matrix is a key concept in linear algebra, characterized by its unique property of being equal to its transpose. In mathematical terms, a matrix \( A \) is symmetric if \( A = A^T \). This implies that the elements of a symmetric matrix are mirrored along its main diagonal.
- If element \( a_{ij} \) is in the \( i^{th} \) row and \( j^{th} \) column, then it holds that \( a_{ij} = a_{ji} \).
- This property ensures that symmetric matrices are always square, meaning they have the same number of rows and columns.
- Symmetric matrices also have real eigenvalues and their eigenvectors can be chosen to be orthogonal.
Matrix Diagonalization
Matrix diagonalization is an important process that transforms a matrix into a simpler form, making computations and theoretical analysis more convenient. A symmetric matrix \( A \) can be expressed in terms of an orthogonal matrix \( Q \) and a diagonal matrix \( D \) as \( A = QDQ^T \). Diagonalization leverages the symmetry of the matrix:
- Here, \( Q \) is an orthogonal matrix, which will be discussed further below, meaning \( Q^T = Q^{-1} \).
- The diagonal matrix \( D \) contains the eigenvalues of \( A \) along its diagonal.
- A significant consequence of diagonalizing a symmetric matrix is that it decomposes the matrix into non-interacting scalar entities in \( D \).
Orthogonal Matrix
An orthogonal matrix is one of remarkable properties that make it highly useful in linear algebra. The defining feature of an orthogonal matrix \( Q \) is that its columns and rows are orthonormal, meaning they are orthogonal to each other and of unit length. This results in:
- The relation \( Q^TQ = QQ^T = I \), where \( I \) denotes the identity matrix.
- Orthogonal matrices preserve lengths and angles, making them analogous to rotations in space.
- Importantly, they simplify complex matrix operations by maintaining numerical stability and preventing error propagation.
Matrix Square Root
The concept of a matrix square root, like its numerical counterpart, involves finding another matrix which, when multiplied by itself, results in the original matrix. For a symmetric positive definite matrix \( A \), a matrix square root exists, meaning there is a matrix \( B \) such that \( B^2 = A \). Key points include:
- A square root matrix \( B \) is also symmetric and positive definite, sharing these properties with \( A \).
- Construction of \( B \) often relies on the diagonalization of \( A \): if \( A = QDQ^T \), the square root can be given by \( B = QD^{1/2}Q^T \), where \( D^{1/2} \) consists of the square roots of the eigenvalues in \( D \).
- Ensuring that \( B \) retains the positive definite attribute of \( A \) is crucial because it means all its eigenvalues are positive, hence maintaining stability and convergence in various applications.
