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The question I'm working on goes like this (from Strang's Linear Algebra 4th edition):

Find the SVD from the eigenvectors $\vec{v}_1, \vec{v}_2$ of $A^TA$ and $A\vec{v}_i=\sigma_i\vec{u}_i$: $$\mathbf{Fibonacci\; matrix}\quad A=\begin{bmatrix}1&1\\1&0\end{bmatrix}.$$

I've calculated $A^TA$ to be $\begin{bmatrix}2&1\\1&1\end{bmatrix}$, and its eigenvalues are $\lambda_1=\frac{3+\sqrt{5}}{2}$ and $\lambda_2=\frac{3-\sqrt{5}}{2}$, which means that the diagonal matrix in the SVD is $\Sigma=\begin{bmatrix}\sqrt{\frac{3+\sqrt{5}}{2}}&0\\0&\sqrt{\frac{3-\sqrt{5}}{2}}\end{bmatrix}$.

And that's where I lose it. My professor rushed through SVD, so I still don't fully understand the process, and the simple examples we did in class don't help much here. Any help would be very much appreciated!

Edit: I've worked out that the unit eigenvectors of $A^TA$ are $\vec{v}_1=\sqrt{\frac{2}{5+\sqrt{5}}}\begin{bmatrix}\frac{2}{1-\sqrt{5}}\\-1\end{bmatrix}$ and $\vec{v}_2=\sqrt{\frac{2}{5-\sqrt{5}}}\begin{bmatrix}\frac{2}{1+\sqrt{5}}\\-1\end{bmatrix}$, but I don't know where to go from here.

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Given an m $\times$ n matrix A, you want to find a factorization A = U Σ V$^*$ such that U is an m $\times$ m unitary matrix, Σ is an m $\times$ n rectangular diagonal matrix with non-negative real numbers on the diagonal, and V$^*$ is the conjugate transpose of an n $\times$ n unitary matrix V.

To clarify the above for a square matrix A, let us continue your example. You already know how to find Σ.

You also have done the work to find V: The columns of V are your unit eigenvectors—in the first column, put the unit eigenvector corresponding to the eigenvalue you chose for the first diagonal entry of Σ.

Finally, calculate U by solving the factorization equation to get U = AV Σ$^{-1}.$ (Note that the transpose of a unitary matrix is also its inverse.)

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  • $\begingroup$ Thanks for your answer; I appreciate that you didn't just give me the answer and tried instead to guide me toward it on my own. I ended up solving that problem a few days after posting the question, however, and I didn't know at the time that I was allowed to answer my own questions (or even that that was a feature of the site). What's proper site protocol as to what I should do now? Accept your answer? Post my solution as an answer? Both? Something else entirely? $\endgroup$ Commented Jul 30, 2018 at 23:49
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    $\begingroup$ @RobertHoward If you like, you may post your solution as a second answer to mine. You should accept the answer, if any, that was most helpful in finding your solution; if you post a solution, do not be afraid to accept it. Finally, you should upvote any useful answer except that you are not allowed to upvote yours. $\endgroup$ Commented Jul 31, 2018 at 16:47

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