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Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ where $x_{n,1}=1$ and the other entries are all zero.

Obviously, the nilpotent matrix $B$ admits only 0 as the eigenvalue. The sum $B+X$ is the circulant matrix and it is well-known that its spectrum are just the nth roots of $1$.

Let us consider an upper triangular matrix $A=(a_{i,j})\in M_n(\mathbb{R})$ with

$$\begin{align*} a_{i,i+1}&=1\\ a_{i,j}&=0 ~~ \operatorname{if}~~ i\geq j \\ a_{i,j}&\in\{0,1\} \operatorname{if}~~ i\leq j+2 \end{align*} $$

Q. Are the eigenvalues of $A+X$ distinct?

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ where $x_{n,1}=1$ and the other entries are all zero.

Obviously, the nilpotent matrix $B$ admits only 0 as the eigenvalue. The sum $B+X$ is the circulant matrix and it is well-known that its spectrum are just the nth roots of $1$.

Let us consider an upper triangular matrix $A=(a_{i,j})\in M_n(\mathbb{R})$ with

$$\begin{align*} a_{i,i+1}&=1& \\ a_{i,j}&=0 ~~ &~~ j\leq i \\ a_{i,j}&\in\{0,1\} &~~ i+2\leq j \end{align*} $$

Q. Are the eigenvalues of $A+X$ distinct?

Let us consider an upper triangular matrix $A=(a_{i,j})\in M_n(\mathbb{R})$ with $a_{kk}=0$ for $k=1,\cdots,n$. Thus $A$ is nilpotent i.e, $A^n=0$. Let us consider $E=(x_{ij})\in M_n(\mathbb{R})$ where $x_{n,1}=1$ and the other entries are all zero.

Q. Are the eigenvalues of $A+E$ distinct?

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ where $x_{n,1}=1$ and the other entries are all zero.

Obviously, the nilpotent matrix $B$ admits only 0 as the eigenvalue. The sum $B+X$ is the circulant matrix and it is well-known that its spectrum are just the nth roots of $1$.

Let us consider an upper triangular matrix $A=(a_{i,j})\in M_n(\mathbb{R})$ with

$$\begin{align*} a_{i,i+1}&=1\\ a_{i,j}&=0 ~~ \operatorname{if}~~ i\geq j \\ a_{i,j}&\in\{0,1\} \operatorname{if}~~ i\leq j+2 \end{align*} $$

Q. Are the eigenvalues of $A+X$ distinct?

Let us consider an upper triangular matrix $A=(a_{i,j})\in M_n(\mathbb{R})$ with $a_{kk}=0$ for $k=1,\cdots,n$. Let us consider $E=(x_{ij})\in M_n(\mathbb{R})$ where $x_{n,1}=1$ and the other entries are all zero.

Q. Are the eigenvalues of $A+E$ distinct?

Let us consider an upper triangular matrix $A=(a_{i,j})\in M_n(\mathbb{R})$ with $a_{kk}=0$ for $k=1,\cdots,n$. Thus $A$ is nilpotent i.e, $A^n=0$. Let us consider $E=(x_{ij})\in M_n(\mathbb{R})$ where $x_{n,1}=1$ and the other entries are all zero.

Q. Are the eigenvalues of $A+E$ distinct?