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Chapter 9: Problem 7

Suppose \(A\) is normal. Relate the polar factors of \(e^{A}\) to \(S=\left(A-A^{T}\right) / 2\) and \(T=\left(A+A^{T}\right) / 2\).

Short Answer

Expert verified
The polar decomposition of \(e^{A}\) is given by \(U = e^{S}\) and \(P = e^{T}\), assuming \([T,S] = 0\).

Step by step solution

01

Express Matrix A in Terms of S and T

First, rewrite the given matrix \(A\) using the symmetric and skew-symmetric components: \[ A = T + S \] where \(T = \frac{A + A^T}{2}\) and \(S = \frac{A - A^T}{2}\). This decomposition splits \(A\) into its symmetric and skew-symmetric parts.
02

Examine the Exponential of A

Consider the exponential of the matrix \(A\): \( e^{A} = e^{T+S} \). Since \(A\) is normal, \(T\) is symmetric and \(S\) is skew-symmetric. The matrix exponential can be expanded as products of exponentials if \(T\) and \(S\) commute, i.e., \([T,S] = 0\). To relate the polar factors, we explore this exponential further.
03

Relate Exponential to Polar Decomposition

The polar decomposition of a matrix \(e^{A}\) is \(e^{A} = UP\), where \(U\) is unitary and \(P\) is positive semi-definite symmetric matrix. Here, the matrix exponential splits naturally into these factors if \([T,S] = 0\): \[ e^{A} = e^{S + T} = e^{S} e^{T} \] where \(e^{S}\) is unitary (since \(S\) is skew-symmetric) and \(e^{T}\) is symmetric positive definite.
04

Conclusion Relating Polar Factors

For the normal matrix \(A = T + S\), where \(T\) and \(S\) commute, the exponential \(e^{A}\) has a polar decomposition such that the polar factors are given by \(U = e^{S}\) and \(P = e^{T}\), assuming the decomposition conditions hold and \(e^{S}\) and \(e^{T}\) are obtained directly as exponential components from \(A\).

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Key Concepts

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

Normal Matrix
A normal matrix is a fundamental concept in linear algebra. It is defined by the equation \(AA^* = A^*A\), where \(A^*\) is the conjugate transpose of \(A\). This commutative property implies that a normal matrix can be diagonalized by a unitary matrix. In simpler terms, you can "rotate" a normal matrix into a diagonal form using complex number rotation without changing its spectral properties.
Here are key points about normal matrices:
  • Diagonalizability: Every normal matrix can be written as \(A = UDU^*\), where \(U\) is unitary and \(D\) is diagonal.
  • Spectral theorem: Normal matrices have eigenvectors that form an orthonormal basis, and their eigenvalues are found on the main diagonal of \(D\).
  • Closure under addition: The sum of two normal matrices is not necessarily normal. However, if they commute, their sum is normal.
Understanding normal matrices is crucial because they allow for simplifications in matrix operations involving powers or exponentials, like the exponential of a normal matrix being closely tied to its polar decomposition.
Skew-symmetric Matrix
A skew-symmetric matrix is a type of square matrix whose transpose equals its negative. In mathematical terms, for a skew-symmetric matrix \(S\), we have that \(S^T = -S\). This property leads to interesting characteristics and implications for such matrices.
Some essential properties of skew-symmetric matrices are:
  • Zero Diagonal: All elements on the diagonal of a skew-symmetric matrix are zero, as each diagonal element must be its own negative.
  • Real Eigenvalues: All eigenvalues of a real skew-symmetric matrix are pure imaginary or zero.
  • Matrix Exponential: The exponential of a real skew-symmetric matrix is a special orthogonal matrix, which is a type of unitary matrix.
In the context of polar decomposition, a skew-symmetric matrix, when exponentiated, results in a unitary (or orthogonal for real matrices), thus offering elegance and simplicity in various numerical and analytical applications.
Symmetric Matrix
A symmetric matrix is characterized by its symmetry across the main diagonal, meaning \(A = A^T\). This makes symmetric matrices not only computationally appealing but also rich in theory.
Key features of symmetric matrices include:
  • Real-Valued Eigenvalues: All eigenvalues of a symmetric matrix are real numbers. This is a significant simplification in many mathematical contexts because it does not involve complex components.
  • Diagonalization with Orthogonal Matrices: A symmetric matrix can be diagonalized with a real orthogonal matrix \(Q\), meaning \(A = QDQ^T\), where \(D\) is a diagonal matrix of eigenvalues.
  • Positive Semi-definite: If all eigenvalues of a symmetric matrix are non-negative, it is called positive semi-definite.
In relation to polar decomposition, symmetric matrices play a crucial role as they often represent the positive semi-definite matrix \(P\) in the decomposition. Thus, understanding symmetric matrices can help analyze polar decompositions as they provide clarity and structure to matrix exponentials and spectral properties.

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Most popular questions from this chapter

Let \(A\) by an \(n\)-by-n symmetric positive definite matrix. (a) Show that there exists a unique symmetric positive definite \(X\) such that \(A=X^{2}\). (b) Show that if \(X_{0}=I\) and $$ X_{k+1}=\left(X_{k}+A X_{k}^{-1}\right) / 2 $$ then \(X_{k} \rightarrow \sqrt{A}\) quadratically where \(\sqrt{A}\) denotes the matrix \(X\) in part (a).

Suppose \(A=X \operatorname{diag}\left(\lambda_{i}\right) X^{-1}\) where \(X=\left|x_{1}\right| \cdots\left|x_{n}\right|\) and \(X^{-1}=\left[\left.y_{1}|\cdots| y_{n}\right|^{H}\right.\), Show that if \(f(A)\) is defined, then $$ f(A)=\sum_{k=1}^{n} f\left(\lambda_{i}\right) x_{i} y_{i}^{H} $$

Suppose $$ A=\left[\begin{array}{cc} \lambda & \mu_{1} \\ \mu_{2} & \lambda \end{array}\right], \quad \mu_{1} \mu_{2}<0 $$ Use the power series definitions to develop closed form expressions for \(\exp (A), \sin (A)\), and \(\cos (A)\).

Give a closed-form expression for \(\log (Q)\) where \(Q\) is a 2-by-2 rotation matrix.

Suppose \(A \in \mathbf{R}^{n \times n}\) has the property that its off-diagonal entries are negative and its column sums are zero. Show that for all \(t, F=\exp (A t)\) has nonnegative entries and unit column sums.
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