# Linked Questions

3answers
17k views

### What are the eigenvalues of matrix that have all elements equal 1? [duplicate]

As in subject: given a matrix $A$ of size $n$ with all elements equal exactly 1. What are the eigenvalues of that matrix ?
3answers
1k views

### Find the eigenvalues of a matrix with ones in the diagonal, and all the other elements equal [duplicate]

Let $A$ be a real $n\times n$ matrix, with ones in the diagonal, and all of the other elements equal to $r$ with $0<r<1$. How can I prove that the eigenvalues of $A$ are $1+(n-1)r$ and $1-r$, ...
3answers
512 views

1answer
3k views

### Eigenvalues of a nxn matrix without calculations [duplicate]

I have a question about the following matrix: $$\begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \\ \end{bmatrix}$$ Find the ...
1answer
1k views

### How to find eigenvectors/eigenvalues of a matrix where each diagonal entry is scalar $d$ and all other entries are $1$ [duplicate]

How would you find eigenvalues/eigenvectors of a $n\times n$ matrix where each diagonal entry is scalar $d$ and all other entries are $1$ ? I am looking for a decomposition but cannot find anything ...
3answers
191 views

### Characteristic polynomial of a matrix 7x7? [duplicate]

Avoiding too many steps, which is the characteristic polynomial of this matrix 7x7? And why? \begin{pmatrix} 5&5&5&5&5&5&5\\5&5&5&5&5&5&5\\5&5&...
3answers
556 views

2answers
210 views

### Eigenvalue decomposition of $A = I - xx^T$ [duplicate]

Let $A = I - xx^T$, where $x \in \mathbb{R}^n$ and $I$ is the identity matrix of $\mathbb{R}^n$ We know that $A$ is a real symmetric matrix, therefore there exists an eigenvalue decomposition of $A$ ...
1answer
396 views

### Minimal polynomial of matrix with rank 1 [duplicate]

Let $A$ be an $n \times n$ matrix over $\mathbb{K}$ such as $\text{rank}(A)=1.$ Show that it's minimal polynomial is $m_A(x) = λ(λ-a) , a \in \mathbb{K}.$ I tried to prove it using induction. My work ...
2answers
116 views

### Compute eigenvalue of a $4\times4$ matrix with all entries $1$ [duplicate]

I have the following matrix, call it A: $\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix}$ I need to find ...
3answers
101 views

### Eigenvector and eigenvalue of $\mathbf A+\lambda\mathbf I$ where $\mathbf A=\mathbf{vv}^\top$ [duplicate]

Given $\mathbf v=\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}$ and $\mathbf A=\mathbf{vv}^\top$, find the eigenvectors and eigenvalues of $\mathbf A+\lambda\mathbf I$. My current work progress ...
3answers
185 views

### Does $\det(I+A) = 1 + \mbox{tr}(A)$ hold if $A$ is a rank-$1$ complex matrix? [duplicate]

If $A$ is a complex $n \times n$ matrix of rank $1$, then $$\det(I+A) = 1 + \mbox{tr}(A)$$ How to approach this problem? Rank-$1$ matrices have special properties. Also, thinking about the ...

15 30 50 per page