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Rodrigo de Azevedo
Rodrigo de Azevedo
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Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.

Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ if $k\geq2$ and $Te_1=e_n$.

Let $A:\mathbb{R}^n\to \mathbb{R}^n$ be the operator given by $Ae_3=Ae_4=e_1$ and $Ae_k=0$ for any other $k$.

It is well-known that the coulmnscolumns of the discrete Fourier matrix are just the eigenvectors of $T$. What about the (analytical proof) eigenvectors of $T+A$?

Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.

Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ if $k\geq2$ and $Te_1=e_n$.

Let $A:\mathbb{R}^n\to \mathbb{R}^n$ be the operator given by $Ae_3=Ae_4=e_1$ and $Ae_k=0$ for any other $k$.

It is well-known that the coulmns of the discrete Fourier matrix are just the eigenvectors of $T$. What about the (analytical proof) eigenvectors of $T+A$?

Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.

Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ if $k\geq2$ and $Te_1=e_n$.

Let $A:\mathbb{R}^n\to \mathbb{R}^n$ be the operator given by $Ae_3=Ae_4=e_1$ and $Ae_k=0$ for any other $k$.

It is well-known that the columns of the discrete Fourier matrix are just the eigenvectors of $T$. What about the (analytical proof) eigenvectors of $T+A$?

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ABB
ABB
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The eigenvectors of adding a particular rank one matrix to the circulant matrix

Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.

Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ if $k\geq2$ and $Te_1=e_n$.

Let $A:\mathbb{R}^n\to \mathbb{R}^n$ be the operator given by $Ae_3=Ae_4=e_1$ and $Ae_k=0$ for any other $k$.

It is well-known that the coulmns of the discrete Fourier matrix are just the eigenvectors of $T$. What about the (analytical proof) eigenvectors of $T+A$?