38
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:
$$X=\left(
\begin{array}{cccc}
0 & 0 & 0 & -...
- 188k
27
votes
Is there a fast way to check if a matrix has any small eigenvalues?
Are you interested in a clever algorithm, or do you just want to get the answer fast?
If the latter, then I would suggest the following:
Use an established off-the-shelf eigenvalue solver.
Vectorize ...
- 1,160
23
votes
Accepted
Existence of double eigenvalue
The answer is 'no'. The generic pair $A$ and $B$ of $4$-by-$4$ Hermitian symmetric matrices will not have any nonzero real linear combination that has a double eigenvalue.
For a specific example, ...
- 108k
23
votes
The statement that $A \ge B$ implies $A^{-1} \le B^{-1}$ is still true for matrices?
This is a well-known fact. A simple proof : setting $y=B^{1/2}x$, we have $\|y\|^2\le y^TB^{-1/2}AB^{-1/2}y$, that is $I_n\le B^{-1/2}AB^{-1/2}$. The eigenvalues of the latter symmetric matrix are ...
- 52.3k
20
votes
Differentiability of Eigenvalues - Perturbation Theory
The complete reference is Kato's book Perturbation theory .... But perhaps you need only the most basic results. Then see my book Matrices (Springer GTM #216), 2nd edition. This is Section 5.2.
Mind ...
- 52.3k
20
votes
Accepted
Eigenvalue pattern
The explanation is pretty simple with a suitable change of basis.
Letting
$$B =
\begin{pmatrix}
1 & 0 & 1 & 0 \\
i & 0 & -i & 0 \\
0 & 1 & 0 & 1 \\
0 & i &...
- 3,446
20
votes
Is there a fast way to check if a matrix has any small eigenvalues?
I am not sure if this is going to be faster than what you are doing now (which is already a clever method), but you can try the following.
Compute $B = A^{-1}$. We know that $A$ has an eigenvalue in $...
- 20.2k
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 $(x-1)^6 (x-\frac92)$. ...
- 79.9k
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 ...
- 38k
17
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 !...
- 52.3k
16
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 ...
- 37.7k
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 ...
- 2,803
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 $\pm\lambda$.
- 52.3k
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 ...
- 52.3k
15
votes
Accepted
Imaginary eigenvalues
Define the unitary and Hermitian matrices
$$U=\left(
\begin{array}{cccc}
0 & 0 & -i & 0 \\
0 & 0 & 0 & -i \\
i & 0 & 0 & 0 \\
0 & i & 0 & 0 \\
\end{...
- 188k
14
votes
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well
Counterexample: Consider the entire function $$ A(z) = \pmatrix{e^z & 0\cr
z & 1\cr}$$
An entire logarithm of $A(z)$ must have eigenvalues $z + 2\pi i n$ and $2 \pi ...
- 54.2k
14
votes
Is there a fast way to check if a matrix has any small eigenvalues?
A symmetric matrix $A$ has an eigenvalue in $(-1,1)$ iff $A^2$ has an eigenvalue in $[0,1]$. Equivalently this means that
$$
\min_{\Vert v\Vert=1} \frac{\Vert Av\Vert^2}{\Vert v\Vert^2} <1.
$$
...
- 34.7k
13
votes
Eigenvalue perturbation theory via Feynman diagrams
I think the simplest way is to use the very simple and very useful resolvant formula
$$ (A+B)^{-1}=A^{-1}-A^{-1}B(A+B)^{-1}.$$
For perturbative theory, we just iterate this formula
$$ (A+B)^{-1}=A^{-...
- 4,361
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. ...
- 2,600
13
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 ...
- 188k
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 ...
- 52.3k
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
$$U=U^{-1}=\left(
\begin{array}{cccc}
0 & 1 & 0 &...
- 188k
13
votes
Accepted
Is the eigenvalue map open?
Yes. Write $D(\lambda_1, \ldots, \lambda_n)$ as the diagonal matrix with diagonal entries $(\lambda_1, \ldots, \lambda_n)$.
Let $A$ be any complex $n \times n$ matrix. We can upper-triangularize $A$ ...
- 156k
12
votes
Accepted
Finding the nearest matrix with real eigenvalues
I have no idea what is going on, but your conjecture is not correct. This is more transparent perhaps in the complex version. Consider
$$
A = \begin{pmatrix} i & a \\ 0&-i \end{pmatrix} , \...
- 24.2k
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 $I=\langle aba\rangle$...
- 26.5k
12
votes
Accepted
Has Nambu's notion of an "eigenoperator" found a place in the mathematical literature?
Such an $X$ is an eigenvector of $\,\operatorname{ad}(H)$. Joint eigenspace decompositions of several $\operatorname{ad}(H_i)$ are commonplace in math since the work of Lie, Killing, Cartan, with the ...
- 31.5k
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)!} \...
- 43.2k
11
votes
Differentiability of eigenvalues of positive-definite symmetric matrices
Let us consider functions $A$ from (an open interval in) $\mathbb{R}$ into the set of symmetric real $n\times n$ matrices (Hermitian complex $n\times n$ matrices behave analogously).
If $A$ is given ...
- 648
11
votes
Accepted
Formula expressing symmetric polynomials of eigenvalues as sum of determinants
This is not a reference, but a short proof.
We use the following (probably known, but see later) lemma on representing a symmetric tensor as a linear combination of rank-1 symmetric tensors.
Lemma. ...
- 109k
11
votes
Formula expressing symmetric polynomials of eigenvalues as sum of determinants
Concerning the reference request:
Several text books [1,2] give the theorem and proof for elementary symmetric polynomials $s_k=$ sum of all $k\times k$ principal minors of the $n\times n$ matrix. ...
- 188k
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