# Tag Info

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### Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

This is true. Indeed, you can estimate the sum of all $n^2$ elements of $A$ rather than individual elements. (Thanks to thomashennecke for observing this, my original answer dealt with the row sums of ...

### Differentiability of Eigenvalues - Perturbation Theory

The complete reference is Kato's book Perturbation theory .... But perhaps you need only the most basic results. Then see my book Matrices (Springer GTM #216), 2nd edition. This is Section 5.2. Mind ...
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### Showing that a certain matrix is not positive definite

Counterexample: let $k=7$, and let $B$ be the circulant matrix with $B_{ij}=1$ iff $i-j \in \{1,2,4\} \bmod 7$. Then $X_B$ is $I + \frac12 J$, with characteristic polynomial $(x-1)^6 (x-\frac92)$. ...
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### Differentiability of eigenvalues of positive-definite symmetric matrices

In the open subset of $M_n(\mathbb{R})$ where the $\lambda_i$ are distinct, they are $C^{\infty}$ functions: this follows from the implicit function theorem. On the other hand, when some eigenvalue ...

### Differentiability of eigenvalues of positive-definite symmetric matrices

The keyword is the Cartan decomposition in the theory of symmetric spaces. In short, when an eigenvalue is simple (its multiplicity is $1$) it is locally an analytic function. But when the ...

### Spectral symmetry of a certain structured matrix

An equivalent trick : Let $J:= \operatorname{diag}(1,i,-1,-i)$. Then $J^*AJ=iB$ where $B$ is real and skew-symmetric. Hence the spectrum of $iB$ (thus that of $A$) comes by pairs $\pm\lambda$.
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### Counting eigenvalues without diagonalizing a matrix

Here is an efficient method. First of all, I must quote that diagonalizing $M$ is not a method, because there is no explicit way to carry this out. It amounts to calculating the roots of a polynomial !...
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### What happens to eigenvalues when edges are removed?

The smallest eigenvalue can go up or down when an edge is removed. For "down": $G=K_n$ for $n\ge 3$. For "up": Take $K_n$ for $n\ge 1$ and append a new vertex attached to a single vertex of the ...
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### Eigenvalues of the complement of a graph

Edit (bis). There are two answers, depending on whether loops about vertices are allowed or not. In addition, the case of regular graphs is completely described. If loops are allowed The relation ...
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### Does there exist a nonsingular graph for which the determinant of its adjacency matrix remains the same upon deleting a vertex?

There exists a simple graph $G=(V,E)$ together with a vertex $v\in V$ such that $\operatorname{det}(A(G))$ and $\operatorname{det}\left(A(G\backslash\{v\})\right)$ are equal and non-zero. Example. ...

### Differentiability of eigenvalues of positive-definite symmetric matrices

As mentionned by other answers, simple eigenvalues are $C^\infty$, while non-simple ones are not. Let me add however two important properties which you can find in Kato's book Perturbation theory of ...
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### Finding the nearest matrix with real eigenvalues

I have no idea what is going on, but your conjecture is not correct. This is more transparent perhaps in the complex version. Consider  A = \begin{pmatrix} i & a \\ 0&-i \end{pmatrix} , \...
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### What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?

The $2^n\times 2^n$ dimensional Hadamard matrices $H_{2^n}$ are also called Sylvester matrices or Walsh matrices. There are only two distinct eigenvalues $\pm 2^{n/2}$, so the eigenvectors are not in ...

### Are eigenvalues preserved under derived equivalence?

Let $A$ be the Nakayama algebra with Kupisch series [3,4], that is $A$ has quiver with two points 1 and 2 and an arrow $a$ from 1 to 2 and an arrow $b$ from 2 to 1 with relations $I=\langle aba\rangle$...

### Largest eigenvalue of the sum of Hermitian matrices

If $A$ has eigenvalues $\lambda_1 \geq \dots \geq \lambda_n$ and $B$ has eigenvalues $\mu_1 \geq \dots \geq \mu_n$, then the largest eigenvalue of $A+B$ can take any value between $\lambda_1+\mu_1$ (...