Linked Questions

33 votes
9 answers
9k views

Determinant of a specially structured matrix ($a$'s on the diagonal, all other entries equal to $b$) [duplicate]

I have the following $n\times n$ matrix: $$A=\begin{bmatrix} a & b & \ldots & b\\ b & a & \ldots & b\\ \vdots & \vdots & \ddots & \vdots\\ b & b & \ldots &...
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  • 5,136
16 votes
8 answers
9k views

How to calculate the determinant of all-ones matrix minus the identity? [duplicate]

How do I calculate the determinant of the following $n\times n$ matrices $$\begin {bmatrix} 0 & 1 & \ldots & 1 \\ 1 & 0 & \ldots & 1 \\ \vdots & \vdots & \ddots & \...
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  • 13.9k
15 votes
3 answers
37k 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 ?
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3 votes
3 answers
5k 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 $1-r$, ...
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  • 51
6 votes
3 answers
843 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 &...
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12 votes
3 answers
575 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&...
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  • 29.5k
4 votes
1 answer
5k 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 ...
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  • 87
4 votes
2 answers
2k 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 $A$ ...
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  • 10.6k
1 vote
1 answer
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 ...
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  • 3,099
4 votes
3 answers
368 views

Characteristic polynomial of a $7 \times 7$ matrix whose entries are $5$ [duplicate]

Avoiding too many steps, what is the characteristic polynomial of the following $7 \times 7$ matrix? And why? \begin{pmatrix}5&5&5&5&5&5&5\\5&5&5&5&5&5&...
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1 vote
3 answers
1k 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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  • 5,544
1 vote
1 answer
2k 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 &...
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  • 293
2 votes
3 answers
914 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,...
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0 votes
3 answers
1k views

To find eigenvalues of matrix with all same element [duplicate]

How many distinct eigenvalues are there in the matrix. $$ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 &...
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  • 125
0 votes
1 answer
1k 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 ...
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  • 409

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