https://file.aiquickdraw.com/mathbot/file/1726850819885uyqoa3vo.jpg

The question in the image is asking whether the following statement is **True** or **False**:

"Let \( A \) be an \( n \times n \) orthogonal matrix. Then, the Frobenius norm \( \| A \|_F = \sqrt{n} \)."

### Explanation:

The Frobenius norm \( \| A \|_F \) of a matrix \( A \) is defined as the square root of the sum of the squares of all the elements of the matrix. If \( A \) is an orthogonal matrix, its rows (and columns) are orthonormal, meaning each row (or column) has a length of 1, and the dot product between different rows (or columns) is zero.

- For an \( n \times n \) orthogonal matrix \( A \), each row has norm 1, meaning the sum of the squares of each element in the row equals 1. Since there are \( n \) rows, the total sum of the squares of all the elements is \( n \).

Therefore, the Frobenius norm of an \( n \times n \) orthogonal matrix \( A \) is:

\[
\| A \|_F = \sqrt{\text{sum of squares of all elements}} = \sqrt{n}.
\]

### Conclusion:

The statement is **True**. \( \| A \|_F = \sqrt{n} \) for an orthogonal matrix \( A \).

---

Would you like a further breakdown of this concept or have any questions?

Here are 5 related questions:
1. What is the definition of an orthogonal matrix?
2. How is the Frobenius norm different from other matrix norms?
3. Can an orthogonal matrix have complex entries?
4. What is the relationship between the determinant of an orthogonal matrix and its properties?
5. How does the Frobenius norm relate to the spectral norm?

**Tip:** The Frobenius norm is often used because it's computationally simpler than some other norms like the spectral norm but still retains geometric significance.

Let A be an n × n orthogonal matrix, then ||A||F = √n.

Explore the properties of orthogonal matrices and their Frobenius norm. Learn why the Frobenius norm of an n × n orthogonal matrix equals √n through a step-by-step explanation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation