True OR False And Justify ихи Let MER nxn as For v(I), is ucis for ģEI lin] V symmetric positive definite matrix ,V(²), vcns Ern Fo are matrix matrix J that the ith alumn is written (M= [ UI), V(2), ,vcny ]). QE Rnan and + { MTQM is diagonal and Q-Conjuate. n vectors then the Positive definite,
True OR False And Justify ихи Let MER nxn as For v(I), is ucis for ģEI lin] V symmetric positive definite matrix ,V(²), vcns Ern Fo are matrix matrix J that the ith alumn is written (M= [ UI), V(2), ,vcny ]). QE Rnan and + { MTQM is diagonal and Q-Conjuate. n vectors then the Positive definite,
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**True or False - Justify Your Answer**
**Problem Statement:**
Let \( M \in \mathbb{R}^{n \times n} \) be a matrix such that the \( i \)-th column is written as \( v^{(i)} \) for \( i \in [1, n] \). In other words, \( M = [v^{(1)}, v^{(2)}, \ldots, v^{(n)}] \).
For a symmetric positive definite matrix \( Q \in \mathbb{R}^{n \times n} \) and \( n \) vectors \( v^{(1)}, v^{(2)}, \ldots, v^{(n)} \in \mathbb{R}^n \) are \( Q \)-conjugate, then the matrix \( M^T Q M \) is diagonal and positive definite.
**Solution Explanation:**
To solve this problem, one would need to confirm whether the statement is true or false and provide a detailed mathematical justification for the answer. This involves verifying the properties of the matrix \( M^T Q M \) and connecting these properties to the given mathematical conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e71c6f0-fdac-4001-9095-2485c121cf1d%2F4d4e7273-0b50-4f17-a32b-567b1bc01924%2F791t4ij_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**True or False - Justify Your Answer**
**Problem Statement:**
Let \( M \in \mathbb{R}^{n \times n} \) be a matrix such that the \( i \)-th column is written as \( v^{(i)} \) for \( i \in [1, n] \). In other words, \( M = [v^{(1)}, v^{(2)}, \ldots, v^{(n)}] \).
For a symmetric positive definite matrix \( Q \in \mathbb{R}^{n \times n} \) and \( n \) vectors \( v^{(1)}, v^{(2)}, \ldots, v^{(n)} \in \mathbb{R}^n \) are \( Q \)-conjugate, then the matrix \( M^T Q M \) is diagonal and positive definite.
**Solution Explanation:**
To solve this problem, one would need to confirm whether the statement is true or false and provide a detailed mathematical justification for the answer. This involves verifying the properties of the matrix \( M^T Q M \) and connecting these properties to the given mathematical conditions.
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