37
votes
Accepted
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:
$$X=\left(
\begin{array}{cccc}
0 & 0 & 0 & -...
- 149k
23
votes
Accepted
How are eigenvalues and eigenvectors affected by adding the all-ones matrix?
This is a special case of a rank one perturbation or a rank one update, and there is plenty of work on such. See the nice 2010 lecture notes by Andre Ran.
- 93.8k
23
votes
Accepted
Existence of double eigenvalue
The answer is 'no'. The generic pair $A$ and $B$ of $4$-by-$4$ Hermitian symmetric matrices will not have any nonzero real linear combination that has a double eigenvalue.
For a specific example, ...
- 98.1k
21
votes
The statement that $A \ge B$ implies $A^{-1} \le B^{-1}$ is still true for matrices?
This is a well-known fact. A simple proof : setting $y=B^{1/2}x$, we have $\|y\|^2\le y^TB^{-1/2}AB^{-1/2}y$, that is $I_n\le B^{-1/2}AB^{-1/2}$. The eigenvalues of the latter symmetric matrix are ...
- 47.5k
20
votes
Accepted
Eigenvalue pattern
The explanation is pretty simple with a suitable change of basis.
Letting
$$B =
\begin{pmatrix}
1 & 0 & 1 & 0 \\
i & 0 & -i & 0 \\
0 & 1 & 0 & 1 \\
0 & i &...
- 1,835
19
votes
Accepted
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 ...
- 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 ...
- 47.5k
17
votes
Accepted
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 $(x-1)^6 (x-\frac92)$. ...
- 71.6k
17
votes
Accepted
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 ...
- 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 ...
- 2,590
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 $\pm\lambda$.
- 47.5k
16
votes
Accepted
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 !...
- 47.5k
15
votes
Accepted
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 ...
- 34.1k
15
votes
Accepted
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 ...
- 47.5k
15
votes
Accepted
Imaginary eigenvalues
Define the unitary and Hermitian matrices
$$U=\left(
\begin{array}{cccc}
0 & 0 & -i & 0 \\
0 & 0 & 0 & -i \\
i & 0 & 0 & 0 \\
0 & i & 0 & 0 \\
\end{...
- 149k
13
votes
Accepted
Why are 1 and -1 eigenvalues of this matrix?
First, one should conjugate all matrices by
$$
\begin{pmatrix}
\operatorname{diag}(\omega_1,\dots,\omega_n) & 0 \\ 0 & 1
\end{pmatrix}
$$
as this converts $S(t)$ to a rotation matrix while ...
- 91k
13
votes
Eigenvalue perturbation theory via Feynman diagrams
I think the simplest way is to use the very simple and very useful resolvant formula
$$ (A+B)^{-1}=A^{-1}-A^{-1}B(A+B)^{-1}.$$
For perturbative theory, we just iterate this formula
$$ (A+B)^{-1}=A^{-...
- 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. ...
- 2,519
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 ...
- 47.5k
13
votes
Accepted
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
$$U=U^{-1}=\left(
\begin{array}{cccc}
0 & 1 & 0 &...
- 149k
12
votes
How are eigenvalues and eigenvectors affected by adding the all-ones matrix?
A cute fact that is trivial to prove is this: define the characteristic polynomial of a matrix $M$ by $\phi_M(x) = |xI-M|$. Then for any $A$ and any $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}(...
- 34.1k
12
votes
Accepted
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} , \...
- 16.7k
12
votes
Accepted
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 ...
- 149k
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 $I=\langle aba\rangle$...
- 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 $\lambda_1+\mu_1$ (...
- 91k
11
votes
Accepted
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 ...
- 51.6k
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)!} \...
- 38.4k
11
votes
Accepted
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-...
- 149k
10
votes
Accepted
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 ...
- 675
Only top scored, non community-wiki answers of a minimum length are eligible