While reading a section on orthogonal matrices, where $\vec{u}_1, ..., \vec{u}_n$ is an orthonormal set of vectors, I find the equation

$$(\vec{u}_1\vec{u}_1^t + ... + \vec{u}_n\vec{u}_n^t)\vec{x}= \begin{bmatrix} | & ... & | \\ \vec{u}_1 & ... & \vec{u}_n \\ | & ... & | \end{bmatrix}\begin{bmatrix} -\vec{u}_1^t-\\ ...\\ -\vec{u}_n^t-\end{bmatrix}\vec{x}$$

Now I'm familiar with summarizing a linear combination of vectors by a matrix-vector product, but in doing so I use my concept of the dot product between a row vector in the matrix and the column vector written to the right. This is related to how, in the matrix product, you're basically doing a bunch of dot-products between rows in the left matrix and columns in the right.

Here is something different though, and I'm not sure how I ought to think about it. The columns are written in the left matrix and rows in the right. I don't see how this is consistent with the dot-product view of matrix products. I could chase indices by thinking of the coordinate-wise definition of matrix products, but I feel like the author intends for a better more natural way of understanding this equation.

So is there a good column-row understanding of matrix products, which makes the above equation sensible? Or is the only good way of understanding this by thinking in terms of the coordinate-wise definition of matrix products?


This can be written as $$ \pmatrix{ \vec u_1 & \cdots & \vec u_n} \left[ \pmatrix{ \vec u_1^T \\ \vdots \\ \vec u_n^T} \vec x \right], $$ where the product in brackets are dot products of $x$ and vectors $u_i$. The resulting numbers are then used to get a linear combination of the vectors $u_i$.

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