What is the name of this matrix?

Question

I have a vector $a=[a_1 \space a_2 \space a_3 \space a_4 \space a_5 \space \cdots a_n]$ and I want to generate following matrix 'A' from it.

$$A=\begin{bmatrix}a_1 & a_2 &a_3\\a_2 & a_3 & a_4\\a_3 & a_4 & a_5\\\vdots & \vdots & \vdots\\a_{n-2} & a_{n-1} & a_n\end{bmatrix}$$

I am looking for a way that can make this vector-to-matrix transformation possible. Secondly, since reverse diagonal elements are same in A, is there a special name for A. Its reverse diagonal elements are same.

Answer 1

It’s not going to be a particularly simple expression, but you can generate $A$ from $a$ with a sequence of matrix operations. I’ll describe the building blocks here and let you work out the specific operations for what you’re trying to do on your own.

1. Left-multiplying a matrix by the $k$th row of the identity matrix picks out the $k$th row of that matrix. If it’s a column vector, this will pick out the $k$th element.
2. Right-multiplying by the $k$th column of the identity picks out the $k$th column.
3. Right-multiplying a column vector with $m$ rows by a row vector with $n$ produces an $m\times n$ matrix.

From the first point it follows that left-multiplying by a permutation of the identity will apply the same permutation to the rows of the matrix. Similarly, right-multiplying by a permutation of the identity will permute the columns.

To trim off the trailing $k$ elements of column vector, the first point tells us that we should multiply by the identity matrix with the last $k$ rows deleted.

Applying the second and third points, we can find that multiplying a column vector $v$ by the row vector $(1,0,0)$ produces the matrix $(v,0,0)$, i.e., the matrix with $v$ as its first column and zeros everywhere else.

To get you started, the matrix that rotates the elements of an $n$-element column vector down one slot would be $$ R=\pmatrix{ 0&1&0&\cdots&0 \\ 0&0&1&\cdots&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots \\ 1&0&0&\cdots&0 }, $$ i.e., $I$ with its rows rotated down by one.