52 votes
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Is there an explicit formula for the hessian of "Determinant"?

The formula you're looking for can be obtained by differentiating Jacobi's formula $$ \frac{\mathrm{d}}{\mathrm{d}t} \det A(t) = \det A(t) \cdot \operatorname{tr}\left( A^{-1} \frac{\mathrm{d}A}{\...
David Zhang's user avatar
  • 1,292
33 votes
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How to prove positivity of determinant for these matrices?

At first, we prove that the determinant is non-zero, in other words, the matrix is non-singular. Assume the contrary, then by the linear dependency of the columns there exist real numbers $\lambda_1,\...
Fedor Petrov's user avatar
30 votes

Is the determinant equal to a determinant?

I will work over a field of characteristic $0$ so that reductive algebraic groups are linearly reductive; presumably there is a way to eliminate this hypothesis. In this case, for an integer $n>1$ ...
30 votes
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Is the determinant equal to a determinant?

I think a result of Hochster allows to get a quick proof that it is not possible to express the determinant of the generic $d \times d$ matrix as the determinant of a $k \times k$ matrices with ...
Libli's user avatar
  • 7,210
30 votes
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How to prove the determinant of a Hilbert-like matrix with parameter is non-zero

I think the reference "Advanced Determinant Calculus" has a pointer to the answer. But I'll still elaborate for it is ingenious. Suppose $x_i$'s and $y_j$'s, $1\leq i,j \leq N$, are numbers such ...
DSM's user avatar
  • 1,196
28 votes
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A Putnam problem with a twist

$\newcommand{\QQ}{\mathbb{Q}} \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\abs}[1]{\left| #1 \right|} \newcommand{\tup}[1]{\left( #1 \right)} \newcommand{\ive}[1]{\left[ #1 \right]} \...
darij grinberg's user avatar
27 votes

Possible values of the determinant for matrices with elements $\{1, 0, -1\}$

Here's a large subset of possible determinants, which may be enough for your application. With $n\times n$ matrices, one can achieve any integral determinant in the interval $[-2^{n-1},2^{n-1}]$. ...
rikhavshah's user avatar
26 votes
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Vandermonde's remarkably clever notation for determinants

The history of the Vandermonde notation is described, in the context of the Vandermonde determinant, in section 2.1 of A case of mathematical eponymy: the Vandermonde determinant (2010). It seems ...
Carlo Beenakker's user avatar
25 votes

How to prove positivity of determinant for these matrices?

To complement Fedor's answer, here is more explicit proof. Let the original matrix be $G$. Let $D_x :=\text{Diag}(e^{x_1},\ldots,e^{x_n})$. Then, we can write \begin{equation*} G = D_x C D_b,\quad\...
Suvrit's user avatar
  • 28.4k
25 votes

On permanents and determinants of finite groups

I won't discuss the permanent part of the question, but I think the other part can be done easily, even without Galois theory. Let $X = X(G)$ denote the character table of $G$ (rows indexed by complex ...
Geoff Robinson's user avatar
25 votes
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Proof that block matrix has determinant $1$

Write the SVD of $A$, say $A=PDQ^T$ with $D$ diagonal with non-negative entries and $P\in O(n),Q\in O(m)$. Then $\sqrt{I_n + AA^T} = P\sqrt{1+D^2}P^T$ and $\sqrt{I_m+ A^TA} = Q\sqrt{1+D^2}Q^T$. This ...
jlewk's user avatar
  • 1,344
24 votes
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"sinc'n determinant"

Let's look at $a_n=\det\left[\frac{(i-1)!}{(2j-1)!}\binom{i^2-\theta^2}j\right]_{i,j=1}^{n}$. By taking out common factors from rows we can write $$a_n=\left(\prod_{i=1}^n (i-1)!(i^2-\theta^2)\right)\...
Gjergji Zaimi's user avatar
23 votes
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Square root of the determinant line

There is no such isomorphism (at least for $g \geq 9$). In O. Randal-Williams, The Picard group of the moduli space of r-Spin Riemann surfaces. Advances in Mathematics 231 (1) (2012) 482-515. I ...
Oscar Randal-Williams's user avatar
22 votes

Expected value of determinant of simple infinite random matrix

Very nice problem! Let me recall you that the determinant of $n \times n$ matrices with entries in $\{0,1\}$ is related to the one of $n+1 \times n+1$ matrices with entries in $\{-1,+1\}$: replace ...
user39115's user avatar
  • 1,785
22 votes
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Determinants of binary matrices

This question is too hard, because easier questions are already known to be hard. The middle column is A046747 in the OEIS, which is essentially equivalent to A057982, the number of singular {±1}-...
Timothy Chow's user avatar
  • 78.1k
22 votes

Proof that block matrix has determinant $1$

We have $Af(A^TA)=f(AA^T)A$ for any reasonable function $f$, including $f(x)=\sqrt{1+x}$. This suffices to check for $f(x)=x^k$ when it is obvious, then approximate your function by a polynomial.
Fedor Petrov's user avatar
21 votes
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Why does this matrix have zero determinant?

Three vectors $v_1,v_2,v_3$ lie in a hyperplane $H:\alpha x+\beta y+\gamma z+\delta t=0$, in this plane we have $Q(v,v):=(\alpha x+\beta y)^2-(\gamma z+\delta t)^2=0,\forall v\in H$. Thus by ...
Fedor Petrov's user avatar
20 votes
Accepted

A determinantal formula

The case $k=n$ is a consequence of the identity $$\int \det(f_j(s_k))\det(g_j(s_k))\prod_{j=1}^N d\mu(s_j) = N!\ \det\left(\int d\mu(t) f_j(t)g_k(t)\right)$$ which I have seen under the names "...
Marcel's user avatar
  • 2,510
19 votes
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Determinantal symmetry: proof requested: Part I

Use Lindstrom-Gessel-Viennot lemma for points $(-a, - j) $ (the first collection of points) and $(i, b) $ (the second collection), both $i, j$ vary from 0 to $c-1$. We see that the determinant counts ...
Fedor Petrov's user avatar
19 votes
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Catalan determinants in search of a proof: Part II

Here is a combinatorial proof. As my running example, I'll take $a=2$ and $n=3$, so I want to show $$\det \begin{bmatrix} 1 & 3 & 1 \\ 1 & 6 & 5 \\ 1 & 10 & 15 \\ \end{bmatrix}...
David E Speyer's user avatar
18 votes

Jacobi's equality between complementary minors of inverse matrices

Not all has been said about this question that is worth saying -- at the very least, someone could have written down the version without the absolute values; but more importantly, there are various ...
darij grinberg's user avatar
18 votes
Accepted

Prove that the matrix $[\Gamma(\lambda_{i}+\mu_{j})]$ is nonsingular

By Andreieff's identity: $$ {\rm det}(A)=\frac{1}{n!}\int_{(0,\infty)^{n}} {\rm det}[e^{-\frac{t_k}{2}}t_k^{\lambda_i-\frac{1}{2}}]_{1\le i,k\le n}\ \times\ {\rm det}[e^{-\frac{t_k}{2}}t_k^{\mu_j-\...
Abdelmalek Abdesselam's user avatar
17 votes
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On the determinant of a class symmetric matrices

Your guess is correct. If the elements outside the diagonal have absolute values less than $1/(n-1)$, the matrix has 'diagonal dominance', thus it is nonsingular. To make the answer self-contained, ...
Fedor Petrov's user avatar
17 votes

Possible values of the determinant for matrices with elements $\{1, 0, -1\}$

I don't know of an answer to your general question of "what determinants are possible for $\{-1,0,1\}$ matrices", but I can answer "is there a {1,0,−1} matrix of order 6 with the ...
Nathaniel Johnston's user avatar
16 votes
Accepted

Pfaffian equals complex determinant?

The answer is YES in every dimension, up to a sign. Here is the calculation. On the one hand, $\det \tilde A=\det(A^2+I)$ because the blocs commute to each other. Therefore $${\rm Pf}(\tilde A)^2=\det(...
Denis Serre's user avatar
  • 51.5k
16 votes

A Putnam problem with a twist

Darij already gave a great elementary detailed exposition. I wanted to remark that all such kinds of results (including the postscript) are special cases of a classical result that (at least in the ...
Gjergji Zaimi's user avatar
15 votes
Accepted

When does the determinant distribute over addition?

let me assume $A$ is invertible, then you ask when $$\det(1+X)=1+\det X,\;\;X=A^{-1}B $$ so if $X$ has eigenvalues $x_i$, $i=1,2,\ldots n$, you would need $$\prod_{i}(1+x_i)=1+\prod_i x_i$$ basically ...
Carlo Beenakker's user avatar
15 votes
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Lang's Jacobian identity: slicker, elementary proof?

Awesome question! I haven't looked at Lang's paper yet, so I can't comment on whether this will be a different approach, but it is elementary. I will make use of Glynn's determinant formula at some ...
Gjergji Zaimi's user avatar
14 votes
Accepted

Finding the closest matrix to $\text{SO}_n$ with a given determinant

(Basically) Full answer For $s \geq 1$ we always take the matrix with diagonals $s^{1/n}$. For $s < 1$ and we have two possibilities. For $n$ small enough we take (still) the matrix with ...
Tim Carson's user avatar

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