Linked Questions

10
votes
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 ?
3
votes
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 $1-r$, ...
12
votes
3answers
512 views

How to calculate this determinant? [duplicate]

How to calculate this determinant? $$A=\begin{bmatrix}n-1&k&k&k&\ldots& k\\k&n-1&k&k&\ldots &k\\\ldots&\ldots&\ldots &&\ldots\\\\k&k&...
5
votes
3answers
394 views

Determinant of a special $n\times n$ matrix [duplicate]

Compute the determinant of the nun matrix: $$ \begin{pmatrix} 2 & 1 & \ldots & 1 \\ 1 & 2 & \ldots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 &...
4
votes
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 ...
1
vote
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 ...
4
votes
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&...
2
votes
3answers
556 views

Help Determinant Binary Matrix [duplicate]

I was messing around with some matrices and found the following result. Let $A_n$ be the $(2n) \times (2n)$ matrix consisting of elements $$a_{ij} = \begin{cases} 1 & \text{if } (i,j) \leq (n,...
4
votes
5answers
194 views

Find the determinant of a matrix definition [duplicate]

Let $A$ be a matrix that is defined like this: $$A_{ij}=\begin{cases} \alpha, & \text{if i=j} \\ \beta , & \text{if i $\ne$ j} \end{cases} $$ So I realized this matrix looks somehow like ...
1
vote
1answer
777 views

Maximum eigenvalue of a hollow symmetric matrix [duplicate]

Is the maximum eigenvalue (or spectral radius) of the matrix with the following form equalled to row or column sum of the matrix? $$ A=\left( \begin{array}{cccc} 0 & a & ... & a \\ a &...
3
votes
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 $A$ ...
0
votes
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 ...
2
votes
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 ...
2
votes
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 ...
0
votes
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 ...

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