9k views

• 13.9k
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
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
843 views

• 29.5k
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
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
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
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&...
1 vote
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
2k views

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
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

15 30 50 per page