Dot product with vector and its transpose? I'm having trouble with the statement: $$||\textbf{v}||^2=\textbf{v}\cdot\textbf{v}=\textbf{v}^T\textbf{v}$$ taking $\textbf{v}$ as a column vector in an orthogonal matrix. How can you do the dot product of a vector and its transpose? Surely that would be like dotting a $n\times1$ matrix with a $1\times n$ matrix? I thought the dot product was only defined for $n\times 1$ vectors?

The dot product "$\cdot$" is also known as scalar product and is defined as the sum of pairwise multiplication: $$\textbf v\cdot \textbf v = \sum_{i=1}^n\textbf v_i^2$$

The last part of the inequality is a matrix multiplication. Because of their dimension ($1\times n$ and $n\times 1$), the result will be $1\times 1$ but the result is exactly the same.