Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 27

MATH152 April 2017

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Question A 27
The matrix $A$ represents a projection in $\mathbb {R} ^{2}$. It is known that $\lambda _{1}=0$ is one eigenvalue with corresponding eigenvector $v_{1}={\begin{bmatrix}-3&1\end{bmatrix}}^{T}$. Find the matrix $A$.

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 what are the possible eigenvalues of a projection matrix, and what is the relation between the first and 2nd eigenvector.

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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. If we write $A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}$, we are told that it satisfies: ${\begin{bmatrix}a&b\\c&d\end{bmatrix}}{\begin{bmatrix}-3\\1\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}$ which gives the equations: $-3a+b=0,\,\,-3c+d=0$ from which we get that the matrix $A$ has the form $A={\begin{bmatrix}a&3a\\c&3c\end{bmatrix}}$. Using that the eigenvalues of a projection are either zero or one, we can pick the 2nd eigenvector to be one which is orthogonal to the first. Choose $v_{2}={\begin{bmatrix}1\\3\end{bmatrix}}$ with eigenvalue one, so that it satisfies the eigenvalue equation: ${\begin{bmatrix}a&3a\\c&3c\end{bmatrix}}{\begin{bmatrix}1\\3\end{bmatrix}}={\begin{bmatrix}1\\3\end{bmatrix}}$. This gives the equations: $10a=1,\,\,10c=3$. Answer: $\color {blue}A={\begin{bmatrix}{\frac {1}{10}}&{\frac {3}{10}}\\{\frac {3}{10}}&{\frac {9}{10}}\end{bmatrix}}$