**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.

M=31+i1-i5 where M=UP and M is a non singular matrix.

Answered: Find the polar decomposition M = UP of…

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

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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Question

**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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