37 votes
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Spectral symmetry of a certain structured matrix

For real $a,b,c$ and imaginary $d$ the matrix $A$ has chiral symmetry, meaning it anticommutes with a matrix $X$ that squares to the identity: $$X=\left( \begin{array}{cccc} 0 & 0 & 0 & -...
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23 votes
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How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

This is a special case of a rank one perturbation or a rank one update, and there is plenty of work on such. See the nice 2010 lecture notes by Andre Ran.
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  • 93.8k
23 votes
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Existence of double eigenvalue

The answer is 'no'. The generic pair $A$ and $B$ of $4$-by-$4$ Hermitian symmetric matrices will not have any nonzero real linear combination that has a double eigenvalue. For a specific example, ...
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21 votes

The statement that $A \ge B$ implies $A^{-1} \le B^{-1}$ is still true for matrices?

This is a well-known fact. A simple proof : setting $y=B^{1/2}x$, we have $\|y\|^2\le y^TB^{-1/2}AB^{-1/2}y$, that is $I_n\le B^{-1/2}AB^{-1/2}$. The eigenvalues of the latter symmetric matrix are ...
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  • 47.5k
20 votes
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Eigenvalue pattern

The explanation is pretty simple with a suitable change of basis. Letting $$B = \begin{pmatrix} 1 & 0 & 1 & 0 \\ i & 0 & -i & 0 \\ 0 & 1 & 0 & 1 \\ 0 & i &...
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19 votes
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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 ...
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  • 21.4k
18 votes

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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  • 47.5k
17 votes
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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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17 votes
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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 ...
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  • 34.5k
16 votes

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 ...
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16 votes

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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16 votes
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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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  • 47.5k
15 votes
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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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15 votes
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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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  • 47.5k
15 votes
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Imaginary eigenvalues

Define the unitary and Hermitian matrices $$U=\left( \begin{array}{cccc} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & -i \\ i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ \end{...
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13 votes
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Why are 1 and -1 eigenvalues of this matrix?

First, one should conjugate all matrices by $$ \begin{pmatrix} \operatorname{diag}(\omega_1,\dots,\omega_n) & 0 \\ 0 & 1 \end{pmatrix} $$ as this converts $S(t)$ to a rotation matrix while ...
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  • 91k
13 votes

Eigenvalue perturbation theory via Feynman diagrams

I think the simplest way is to use the very simple and very useful resolvant formula $$ (A+B)^{-1}=A^{-1}-A^{-1}B(A+B)^{-1}.$$ For perturbative theory, we just iterate this formula $$ (A+B)^{-1}=A^{-...
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  • 4,266
13 votes

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. ...
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13 votes

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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13 votes
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Eigenvalues come in pairs

This follows from the identities $$A_1^{-1}=UA_1U^{-1},\;\;A_2^{-1}=UA_2U^{-1},$$ $$A_1^{\top}=VA_1V^{-1},\;\;A_2^{\top}=VA_2V^{-1},$$ with $$U=U^{-1}=\left( \begin{array}{cccc} 0 & 1 & 0 &...
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12 votes

How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

A cute fact that is trivial to prove is this: define the characteristic polynomial of a matrix $M$ by $\phi_M(x) = |xI-M|$. Then for any $A$ and any $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}(...
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12 votes
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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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12 votes
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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 ...
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12 votes

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 $I=\langle aba\rangle$...
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  • 22.3k
11 votes

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 $\lambda_1+\mu_1$ (...
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  • 91k
11 votes
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No arbitrary product of matrices has eigenvalue 1?

In the case $n=4$ you could have $D = \pmatrix{0 & 0 & 0 & 1\cr 0 & 0 & 1 & 0\cr 0 & 1 & 0 & 0\cr 1 & 0 & 0 & 0}$, in which case $A_1 A_2 A_3 A_4 A_1 ...
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11 votes

Eigenvalues and eigenvectors of the matrix with entries $\dbinom{n+1}{2j-i}$ for $i, j = 1, 2, \ldots, n$

Here is a proof for your identity in Question 3: define the functions \begin{equation} F(n,k):=\frac{\left(-1\right)^{k-i} \left(k-1\right)! \left(n-k\right)!} {2\left(i-1\right)! \left(n-i\right)!} \...
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11 votes
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Kernel of the Laplacian + a function

Q: Can we conclude that $Lu=\Delta u+ fu=0$ has only zero (or constant solutions) if we assume $f$ non-constant? A: No, a counter example in one dimension is the Mathieu equation, which has non-...
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10 votes
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Linearly constrained eigenvalue problem

I think Pushpendre's answer isn't quite right, but it gets you most of the way there. Getting rid of that pesky constant term is a bit tricky relative to the homogeneous case. Let's take his ...
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