Question

Let \(y\) be a non-zero vector of size \(2022 \times 1\). Which of the following statement(s) is/are TRUE?

• A. \(yy^T\) is a symmetric matrix.
• B. \(y^T y\) is an eigenvalue of \(yy^T\).
• C. \(yy^T\) has a rank of 2022.
• D. \(yy^T\) is invertible.

Hint:

The matrix \(yy^T\) is symmetric, and its eigenvalue is the scalar \(y^T y\), which is the dot product of \(y\) with itself.

Answer 1

The Correct Option is A, B

Let's analyze each statement: - (A) \(yy^T\) is a symmetric matrix: True. The matrix \(yy^T\) is a rank-1 matrix formed by the outer product of a vector with itself. Since \(yy^T\) is always equal to its transpose, it is symmetric. - (B) \(y^T y\) is an eigenvalue of \(yy^T\): True. The scalar \(y^T y\) (which is the dot product of \(y\) with itself) is an eigenvalue of the matrix \(yy^T\). The rank of \(yy^T\) is 1, and it has one non-zero eigenvalue, which is \(y^T y\). - (C) \(yy^T\) has a rank of 2022: False. The rank of \(yy^T\) is 1, as it is the outer product of a single vector with itself. Thus, it cannot have a rank of 2022. - (D) \(yy^T\) is invertible: False. Since \(yy^T\) has rank 1, it is not invertible because it does not have full rank. Thus, the correct answers are (A) and (B).