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Find the polar decomposition M = UP of the matrir M = 1-i 5

Find the polar decomposition M = UP of the matrir M = 1-i 5

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Find the polar decomposition \( M = UP \) of the matrix \( M = \begin{pmatrix} 3 & 1+i \\ 1-i & 5 \end{pmatrix} \). **Explanation:** Polar decomposition is a way to express a matrix as the product of a unitary matrix \( U \) and a positive-semidefinite matrix \( P \). This is similar to how any complex number can be represented in terms of its magnitude and a phase. Here, \( U \) represents the "direction" while \( P \) represents the "magnitude." The given matrix is \( M = \begin{pmatrix} 3 & 1+i \\ 1-i & 5 \end{pmatrix} \), where \( i \) is the imaginary unit, satisfying \( i^2 = -1 \). **Steps to Solve:** 1. Compute the Hermitian part of the matrix, \( M^*M \). 2. Find the positive-semidefinite matrix \( P = (M^*M)^{1/2} \). 3. Determine the unitary matrix \( U = MP^{-1} \). The exact calculations will involve standard matrix operations, including conjugate transpositions, multiplications, and eigenvalue decompositions.
Transcribed Image Text:**Problem Statement:** Find the polar decomposition \( M = UP \) of the matrix \( M = \begin{pmatrix} 3 & 1+i \\ 1-i & 5 \end{pmatrix} \). **Explanation:** Polar decomposition is a way to express a matrix as the product of a unitary matrix \( U \) and a positive-semidefinite matrix \( P \). This is similar to how any complex number can be represented in terms of its magnitude and a phase. Here, \( U \) represents the "direction" while \( P \) represents the "magnitude." The given matrix is \( M = \begin{pmatrix} 3 & 1+i \\ 1-i & 5 \end{pmatrix} \), where \( i \) is the imaginary unit, satisfying \( i^2 = -1 \). **Steps to Solve:** 1. Compute the Hermitian part of the matrix, \( M^*M \). 2. Find the positive-semidefinite matrix \( P = (M^*M)^{1/2} \). 3. Determine the unitary matrix \( U = MP^{-1} \). The exact calculations will involve standard matrix operations, including conjugate transpositions, multiplications, and eigenvalue decompositions.
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