Answered: Show that every square matrix A can be…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Show that every square matrix A can be factored as A = RQ, where R is symmetric, positive semidefinite and Q is orthogonal.

Expert Solution

Step 1

Given: Square matrix, $A$
We need to show that every square matrix, $A$ can be factored as $A = R Q$ where - $R = symmetric+positive semidefinite; Q = Orthogonal$

Step 2

Let's write the single value decomposition of some $n \times n$ matrix $A .$
Since $U$ is orthogonal matrix i.e. $U U^{T} = I$ , we get-
$A = U \sum_{}^{} V^{T} = U \sum_{}^{} I V^{T} = U \sum_{}^{} U^{T} U V^{T} = (U \sum_{}^{} U^{T}) (U V^{T})$
Here, $U \sum_{}^{} U^{T}$ is orthogonally diagonalizable because $U$ is an orthogonal and $\sum_{}^{}$ is a diagonal matrix.
If a matrix is orthogonally diagonalizable, then it is symmetric.
$∴$ $U \sum_{}^{} U^{T}$ is symmetric.

Step by step

Solved in 4 steps

Knowledge Booster

$Relations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions