Questions tagged [spectral-radius]
The spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its spectrum.
335 questions
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Non-trivial lower bound for minimum non-zero determinant among square submatrices
Let $A \in \mathbb{R}^{m, n}$, with $n > m$ and $\text{rank}(A) = m$, and let $\mathcal{B}$ be the set of square $m\times m$ submatrices of $A$.
In other words, $\mathcal{B}$ contains all matrices $...
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Find scalar that minimises spectral radius of a matrix
I have a matrix $A$ that is a quadratic function of a real scalar $\beta$ and real constant matrices $B,C,D$:
$$
A = B + \beta C + \beta^2 D
$$
I want to find the value of $\beta$ that minimises the ...
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Gelfand's spectral radius formula for integral kernel operators and $L^p$ norms
For the sake of concreteness, I will be working on the Hilbert space $H=L^2([0,1])$. Let $T$ be a bounded compact operator on $H$ given by a kernel $K\in L^2([0,1]^2)$. In my application, $K$ is non-...
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Spectral radius of block matrix with positive semidefinite blocks
Let $M$ be the block matrix
$$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}, $$
where $A,B,C,D$ are all real, square, and symmetric positive semidefinite. Is it true that the spectral ...
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Show $(\lambda I-A)^{-1}= \sum_{n=0}^\infty \frac{A^n}{\lambda^{n+1}}$ for $|\lambda|>$ spectral radius of $A$
I have the follow problem from a Chinese functional analysis book. I try to use English to describe it. If there are any mistakes or places that are unclear, please tell me and I will correct.
Let $X$ ...
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Looking for example: non-negative matrix with spectral radius eigenvalue having multiplicity > 1 and another eigenvalue of same modulus
I'm looking for an example of a real square matrix $A \in \mathbb{R}^{n \times n}$ with non-negative entries that satisfies all of the following conditions:
There exists an eigenvalue $\lambda \ne \...
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How to show $|\lambda| > \limsup_{n\to\infty} \|A^n\|^{\frac{1}{n}} \implies (\lambda I - A)^{-1} \in L(X)$?
Let $X$ be a Banach space. Denote
$$
L(X) = \{T:X\rightarrow X \mid T \text{ is linear and bounded} \}.
\tag{1}
$$
Then, for any $A\in L(X)$, why
$$
|\lambda| > \limsup_{n\to\infty} {\|A^n\|}^{\...
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When does $\rho(A) > 1$ for adjacency matrices?
Recently, I’ve been studying the Katz centrality index on sparse graphs, and now I’m shifting toward very dense graphs, following Noferini & Wood$^\color{magenta}{\star}\!$. However, a thought ...
user769611
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Spectral radius of an infinite nonnegative Hessenberg Toeplitz matrix
Let $k, a, b > 0$ and $b < 1$. Define the $n \times n$ Hessenberg Toeplitz matrix
$$ {\bf A}_n := \begin{bmatrix}
ab & ab^2 & ab^3 & \cdots & ab^n \\
...
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Bound for 2-norm of a diagonalizable matrix with spectral radius $\rho{X}$
Question: is the following proof correct?
Let $X \in \mathbb{R}^{n \times n}$ be a diagonalizable matrix, i.e., $X=V D V^{-1}$ for some invertible matrix $V$ and diagonal matrix $D$. Let $\rho(X)=\max ...
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Prove that the Jacobi method converges for every $2 \times 2$ symetric definite positive matrix
Prove that the Jacobi method converges for every $2 \times 2$ system given by a symetrical definite positive matrix.
I tried building the iteration matrix $B_J = -D^{-1} (L+U)$ given a matrix
$$
A=
\...
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Let $A$ and $B$ be $n\times n$ Hermitian matrices. Then $A\geq B$ if and only if $\rho(BA^{-1})\leq 1$
Mostly what the title says. We have further that $A$ is positive definite and $B$ is positive semi definite. Assuming that $A\geq B$, I am able to conclude that:
$$x^*(I-BA^{-1})x = x^*(A-B)A^{-1}x \...
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Bound of Joint Spectral Radius of Matrix plus Identity
Given a Matrix $A$ with spectral radius $\rho(A)$, we can prove that $\rho(A+I)\leq\rho(A)+1$. This is the case, since the spectral radius is defined as $\rho(A)=\max{|\lambda_i|}$ and the eigenvalues ...
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How to prove that $\left\|A^d\right\|\leq C\rho(A)\left\|A\right\|^{d-1}$ using Cayley-Hamilton?
If $A$ is a complex matrix, and $\rho(A)$ is its spectral Radius, then Jairo Bochi's answer to this question,
$$\left\|A^d\right\|\leq C\rho(A)\left\|A\right\|^{d-1};\quad (Eq.\, I)$$
where $C>0$.
...
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Operator norm and trace inequality for blocks of positive matrix
For a complex, Hermitian, positive semi-definite matrix
$ X \in \mathbb{C}^{2d \times 2d}$ of the form
$$ X = \begin{bmatrix} A & B \\ B & C \end{bmatrix} \, ,$$
such that $A, B, C \in \mathbb{...
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