Science:Math Exam Resources/Courses/MATH307/December 2008/Question 06 (b)

MATH307 December 2008

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Question 06 (b)
Let u be a unit vector and Q = I - 2uuT. Show: (b) The matrix Q + iI is invertible.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Think eigenvalues and recall part (a).

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Recall the definition of the eigenvalue, we know that $-i$ is one of Q’s eigenvalues when $det(Q+iI)=0$, and $Q+iI$ is invertible if and only if $det(Q+iI)\neq 0$. From part a, we know that Q is a symmetric matrix. A symmetric matrix will only have real eigenvalues and since $-i$ is not real, it is not an eigenvalue of Q and $det(Q+iI)$ is never equal to zero. Therefore, the matrix $Q+iI$ is invertible.

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