.3.5. For n×n matrices A and B, explain why each of the following in- equalities is valid. (a) trace (B)² ≤n [trace (B*B)]. (b) trace (B²) ≤ trace (BTB) for real matrices. (c) trace (ATB) ≤ trace (ATA) + trace (BTB) 2 for real matrices.
.3.5. For n×n matrices A and B, explain why each of the following in- equalities is valid. (a) trace (B)² ≤n [trace (B*B)]. (b) trace (B²) ≤ trace (BTB) for real matrices. (c) trace (ATB) ≤ trace (ATA) + trace (BTB) 2 for real matrices.
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
![### 5.3.5. Matrix Inequalities
For \( n \times n \) matrices **A** and **B**, explain why each of the following inequalities is valid.
#### (a)
\[ | \text{trace}(\mathbf{B}) |^2 \leq n [ \text{trace}(\mathbf{B}^* \mathbf{B}) ]. \]
#### (b)
\[ \text{trace}(\mathbf{B}^2) \leq \text{trace}(\mathbf{B}^T \mathbf{B}) \text{ for real matrices}. \]
#### (c)
\[ \text{trace}(\mathbf{A}^T \mathbf{B}) \leq \frac{\text{trace}(\mathbf{A}^T \mathbf{A}) + \text{trace}(\mathbf{B}^T \mathbf{B})}{2} \text{ for real matrices}. \]
### Explanation
These inequalities are mathematical expressions involving the trace function, which is the sum of the elements on the main diagonal of a matrix. Each inequality has specific conditions or operations, such as transposition (denoted as \( \mathbf{B}^T \)) and conjugate transpose (denoted as \( \mathbf{B}^* \)).
Understanding these inequalities helps in various mathematical and physical applications, particularly in matrix analysis and quantum mechanics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98b6e310-08ba-4e1d-a9bc-704b45d2ce6c%2F89ca51f6-5043-49a6-868f-51ff8b3bbbc4%2Fdbtmqc9_processed.png&w=3840&q=75)
Transcribed Image Text:### 5.3.5. Matrix Inequalities
For \( n \times n \) matrices **A** and **B**, explain why each of the following inequalities is valid.
#### (a)
\[ | \text{trace}(\mathbf{B}) |^2 \leq n [ \text{trace}(\mathbf{B}^* \mathbf{B}) ]. \]
#### (b)
\[ \text{trace}(\mathbf{B}^2) \leq \text{trace}(\mathbf{B}^T \mathbf{B}) \text{ for real matrices}. \]
#### (c)
\[ \text{trace}(\mathbf{A}^T \mathbf{B}) \leq \frac{\text{trace}(\mathbf{A}^T \mathbf{A}) + \text{trace}(\mathbf{B}^T \mathbf{B})}{2} \text{ for real matrices}. \]
### Explanation
These inequalities are mathematical expressions involving the trace function, which is the sum of the elements on the main diagonal of a matrix. Each inequality has specific conditions or operations, such as transposition (denoted as \( \mathbf{B}^T \)) and conjugate transpose (denoted as \( \mathbf{B}^* \)).
Understanding these inequalities helps in various mathematical and physical applications, particularly in matrix analysis and quantum mechanics.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps with 5 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
