Linked Questions
116 questions linked to/from Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
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:
$$A=\begin{bmatrix} a & b & \ldots & b\\ b & a & \ldots & b\\ \vdots & \vdots & \ddots & \vdots\\ b & b & \ldots &...
- 5,136
16
votes
8
answers
9k
views
How to calculate the determinant of all-ones matrix minus the identity? [duplicate]
How do I calculate the determinant of the following $n\times n$ matrices
$$\begin {bmatrix}
0 & 1 & \ldots & 1 \\
1 & 0 & \ldots & 1 \\
\vdots & \vdots & \ddots & \...
- 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 ?
- 275
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 $1-r$, ...
- 51
6
votes
3
answers
843
views
Determinant of a special $n\times n$ matrix [duplicate]
Compute the determinant of the nun matrix:
$$
\begin{pmatrix}
2 & 1 & \ldots & 1 \\
1 & 2 & \ldots & 1\\
\vdots & \vdots & \ddots & \vdots\\
1 & 1 &...
- 3,646
12
votes
3
answers
575
views
How to calculate this determinant? [duplicate]
How to calculate this determinant?
$$A=\begin{bmatrix}n-1&k&k&k&\ldots& k\\k&n-1&k&k&\ldots &k\\\ldots&\ldots&\ldots &&\ldots\\\\k&k&...
- 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 ...
- 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 $A$ ...
- 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 ...
- 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&...
- 75
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 ...
- 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?
$$
A=\left( \begin{array}{cccc}
0 & a & ... & a \\
a &...
- 293
2
votes
3
answers
914
views
Help Determinant Binary Matrix [duplicate]
I was messing around with some matrices and found the following result.
Let $A_n$ be the $(2n) \times (2n)$ matrix consisting of elements $$a_{ij} = \begin{cases} 1 & \text{if } (i,j) \leq (n,...
user394255
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 &...
- 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 ...
- 409