52
votes

Accepted

### 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}{\...

- 1,292

33
votes

Accepted

### 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,\...

- 93.5k

31
votes

Accepted

### How to prove this determinant is positive?

Here are some ideas how to decide the conjecture. (EDIT: In fact these ideas lead to a proof of the conjecture as Terry Tao explained in two comments below.)
As Christian Remling and Will Sawin ...

- 89.9k

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 $n>1$ ...

Community wiki

30
votes

Accepted

### 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 ...

- 6,701

30
votes

Accepted

### 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 ...

- 1,156

27
votes

Accepted

### 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]}
\...

- 31.8k

26
votes

Accepted

### 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 ...

- 160k

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\...

- 28k

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 ...

- 40.3k

25
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 $[-2^{n-1},2^{n-1}]$.
...

- 361

24
votes

Accepted

### "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)\...

- 83.2k

24
votes

Accepted

### 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 ...

- 1,015

23
votes

Accepted

### 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 ...

- 17.3k

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 ...

- 1,775

21
votes

Accepted

### 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}-...

- 70.6k

21
votes

Accepted

### 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 ...

- 93.5k

21
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.

- 93.5k

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 "...

- 2,440

19
votes

Accepted

### 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 ...

- 93.5k

19
votes

Accepted

### 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}...

- 144k

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-\...

- 20.4k

17
votes

Accepted

### 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, ...

- 93.5k

16
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 ...

- 31.8k

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(...

- 49.2k

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 ...

- 83.2k

16
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 ...

- 5,153

15
votes

Accepted

### Expected size of determinant of $AA^T$ for non-square random $A$

By the Cauchy-Binet theorem, $\det AA^T=\sum (\det B)^2$, where $B$
ranges over all $m\times m$ submatrices of $A$. The expected value of
$(\det B)^2$ is $(m+1)!/4^m$, so the expected value of $\det ...

- 46.3k

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 ...

- 160k

14
votes

### Determinant of a $k \times k$ block matrix

Subtracting the last row of blocks from the first $k-1$ rows of blocks, we obtain
$$\begin{bmatrix}A-B & O & O & \dots & O & B-A\\ O & A-B & O & \dots & O & B-...

- 2,182

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