## New answers tagged linear-algebra

5
votes

Accepted

### When do the nonzero eigenvalues of a directed graph Laplacian have the same absolute value?

This is false.
Run this code in SageMath; you can do this at sagecell.sagemath.org if you do not have SageMath already installed on your own computer.
...

5
votes

### How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$

This does not hold in general.
Note that $\beta=A(A^TA)^{-1}A^Ty\in \mathbb{R}^m$ belongs to the image ${\rm Im} A$ (that is, to the column space of $A$). Also, $A^T\beta=A^TA(A^TA)^{-1}A^Ty=A^Ty$, so ...

1
vote

Accepted

### The set of strongly positive forms is a closed cone

The set $P$ of strongly positive forms is the convex hull of the cone of simple strongly positive forms which will be denoted by
$$\mathcal{C}=\{\Omega\in\bigwedge^{p,p}V|\Omega=\sqrt{-1}^{p^{2}}w\...

2
votes

### Krein-Rutman for integral transforms: proof of convergence to leading eigenvector

The result you are looking for is due to Birkhoff and Hopf. See An elementary proof of the Birkhoff-Hopf theorem by Eveson and Nussbaum which has a readable introduction. The Birkhoff-Hopf theorem ...

9
votes

Accepted

### Efficiently computing $\prod_{i=1}^{n} A_i$

To be unambiguous about the order of multiplication, let $B(n) = A_1 A_2 \cdots A_n$. We have the D-finite recurrences
$B(n)_{r,1} = (\frac{n}{n-1})^k B(n-1)_{r,1} + n^k B(n-2)_{r,1}$
$B(n)_{r,2} = B(...

0
votes

### Optimizing a matrix quadratic form with respect to Loewner order

If you meant that you want that property to hold for all matrices $X$ simultaneously, then the answer is that there exists no trace-1 matrix with that property for $k\geq 2$. Let again $P$ have ...

1
vote

Accepted

### Optimizing a matrix quadratic form with respect to Loewner order

This seems easy so maybe I am misreading some condition.
You may assume without loss of generality that $P^TP = I_k$; otherwise orthonormalize its columns / apply Gram-Schmidt to get a new $P$ with ...

1
vote

### Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?

I found an intuitive explanation that works for me here. It demonstrates that this effect already starts to happen in as few as three dimensions based on the intuition that the equator is the largest ...

2
votes

### Are automorphisms of matrix algebras necessarily determinant preservers?

From an abstract point of view a determinant on any ring $R$ should be defined as a multiplicative polyomial map $N$ from $R$ to some central subring so that each element $a$ of $R$ satisfies ...

3
votes

Accepted

### Derivative norm estimates

The answer by Bazin (https://mathoverflow.net/users/21907/bazin), Faa di Bruno's formula for vector valued functions, URL (version: 2012-09-04): https://mathoverflow.net/q/106339
is providing a ...

8
votes

### On minimal eigenvalue

$\newcommand\Vol{\mathop{\mathrm{Vol}}}\newcommand\tr{\mathop{\mathrm{tr}}}$The estimate on the sum of determinants is rewritten.
Here is the proof that one of the eigenvalues indeed does not exceed $...

15
votes

Accepted

### Almost orthogonal maps $f:\omega \to \{-1,1\}$

Here is a family with cardinality continuum. For a nonnegative integer $n$, let the base $2$ expansion of $n$ be
$$n = \sum_{k=0}^{\infty} b_k(n) 2^k.$$
So $b_k(n) \in \{ 0, 1 \}$ and is equal to $0$ ...

4
votes

Accepted

### $\omega\times\omega$-Hadamard matrices

One example would be the universal orthogonal array from the theory of factorial experimental design. The matrix is presented below, followed by its mathematical description.
$\begin{array}{cccccccc}
—...

1
vote

### Graceful labeling of the complete bipartite graph and its laplacian quadratic form diagonalized

Yes, just take $P=T$. In general, congruence preserves only inertia, but since you want $D$ as a diagonal matrix of eigenvalues of $L$, you need to preserve eigenvalues and not just their signs, so ...

2
votes

### Partial inverse of a matrix - or does it have its own name?

Partial inversion is a valid name, and yeah, it is still a hot topic in S-matrix theory and Pseudo-Unitary Quantum Mechanics.
The partial inversion transformation $\hat{Y}_{ik}$ over a general ...

0
votes

### Is a probabilistic implementation of unitaries invertible?

Suppose the values of a discrete random variable are unitary matrices. I'm not sure I've correctly understood the question, but it looks as if maybe it is whether the expected value of such a random ...

-1
votes

### Vector of integers such that almost all dot products are positive

Geometrically: All permutations of x have the same dot product with (1, 1, 1, 1...) and we have assumed that this dot product is not zero. Thus, all the permutations lie on a cone with axis (1, 1, 1, ....

4
votes

Accepted

### Is a probabilistic implementation of unitaries invertible?

This is not true, even for the case where $A$ is a quantum state. Notation-wise, I will eschew exponentials of $\text{ad}$ and instead write the unitary transformations explicitly. I will also use the ...

9
votes

### Vector of integers such that almost all dot products are positive

Alternatively, you may start with $a_i=-i+t$ where $t$ is chosen so that $\sum a_ix_i=0$. By rearrangement inequality, we have $\sum a_i x_{\pi_i}>0$ for every not-identity permutation $\pi$. Then ...

18
votes

### Are automorphisms of matrix algebras necessarily determinant preservers?

Here is a positive result. Every finite-dimensional algebra $A$ over a field $K$ has an intrinsic determinant, and in fact an intrinsic characteristic polynomial, which is preserved by all ...

11
votes

Accepted

### Vector of integers such that almost all dot products are positive

Sure, this is true. By the comment of Joseph Van Name we basically want a hyperplane passing through $0$ separating all the permutations of our numbers viewed as vectors in $\mathbb{R}^n$ from $(x_1, \...

19
votes

Accepted

### Are automorphisms of matrix algebras necessarily determinant preservers?

Every finite dimensional algebra is a subalgebra of a matrix algebra. Indeed, to write an algebra $A$ as a subalgebra of a matrix algebra is the same as to choose a finite-dimensional faithful $A$-...

1
vote

### Reference request: "Higher order eigentuples" as generalized eigenvectors?

$\newcommand\la\lambda\newcommand\La\Lambda$
Given a square matrix $M$ of size $n\times n$, find a matrix $V$ of size $n\times 2$ and a matrix $\Lambda$ of size $2\times 2$ such that
$$
MV = V\Lambda....

1
vote

### Reference request: "Higher order eigentuples" as generalized eigenvectors?

This seems to be equivalent to finding invariant two dimensional subspaces. One way to algebraically describe two dimensional subspaces is the vector space $V'=\Lambda^2V$ (the second wedge product). ...

2
votes

Accepted

### “Smallest” non-zero linear combination of vectors to obtain a non-negative vector

I will show that $\delta_j\le j-1$. I suspect that in fact $\delta_j=j-1$, but I don't have a complete argument for it and it might be wrong.
Let $L$ be the lattice spanned by the columns of $A$ and ...

5
votes

Accepted

### Reference request: "Higher order eigentuples" as generalized eigenvectors?

The observation that $MV = V\Lambda$ is essentially an invariant subspace relation is commonly used in methods to solve algebraic Riccati equations via Hamiltonian matrices; for instance, the Schur ...

2
votes

### Reference request: "Higher order eigentuples" as generalized eigenvectors?

Let me assume that $M$ is symmetric, then we can work in a basis where it is diagonal, $M=\operatorname{diag}(\mu_1,\mu_2,\ldots \mu_n)$. I denote the vectors $v_i=V_{i1}$ and $u_i=V_{i2}$, $i=1,2\...

1
vote

Accepted

### How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?

Let $v \in \mathbb{R}^d$ and let $W$ be the matrix whose $(i,j)$ entry is $v_i + v_j$. Then the matrix $W$ has two non-trivial eigenvalues which are given by
$$
\left(\sum_i v_i\right) \pm \sqrt{d} \...

1
vote

Accepted

### Non-degeneracy in hyperplane intersections of canonical curves

Edit: my answer contained an incomplete argument, here's a second attempt.
Assume by way of contradiction that for a generic hyperplane the expression $$
\begin{aligned}
D(p_1,\ldots,p_6)=\det (p_1 ...

1
vote

### Maximizing a quadratic form involving a trace-bounded positive definite matrix?

We assume that the dimension of the matrices is $n\geq 2$. Let $Sym$ (resp. $Skew$) denote the sets of symmetric (resp. skew-symmetric) matrices.
i) Up to an orthonormal change of basis, we may assume ...

5
votes

Accepted

### A problem about matrix inverse and regularization methods

There is a lot of research about this. I can recommend the books
Engl, Heinz Werner, Martin Hanke, and Andreas Neubauer. Regularization of inverse problems. Vol. 375. Springer Science & Business ...

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