**3**

votes

**1**answer

1k views

### Determinant and symmetric power

Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power ...

**4**

votes

**3**answers

2k views

### What's the maximum determinant of the (0, 1) matrix from M(n, R)?

If there's no exact formula what's the nearest upper and lower bounds do you know?

**3**

votes

**2**answers

2k views

### Matrix products under which the determinant behaves multiplicatively

The determinant behaves multiplicatively with respect to the usual matrix product
$$
\det(AB) = \det(A)\det(B),
$$
and also with respect to the Kronecker (or tensor) product of square matrices
$$
\...

**3**

votes

**0**answers

936 views

### Determinant of a sum of a diagonal matrix, a dyadic product matrix, and a Hermitian Toeplitz matrix

Hi
From a physics problem, I am trying to evaluate exactly the following kind of determinant:
G = A + M + N.
A is diagonal
M is a product of a column (of 1s) and a row matrix
N is a Hermitian ...

**7**

votes

**1**answer

815 views

### Appropriate journal to publish a determinantal inequality

I have recently made the following observation:
Let $v_i := (v_{i1}, v_{i2})$, $1 \leq i \leq k$, be non-zero positive elements of $\mathbb{Q}^2$ such that no two of them are proportional. Let $M$...

**9**

votes

**2**answers

3k views

### Determinant of a 3x3 Magic Square

Hello guys. This is my first question with mathOverflow so I hope my etiquette is up to par here.
My question is regarding a 3x3 magic square constructed using the la Loubere method (see la Loubere ...

**60**

votes

**19**answers

23k views

### Why were matrix determinants once such a big deal?

I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...

**20**

votes

**6**answers

3k views

### Expressing adj(A) as a polynomial in A?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\mathop{\mathrm{det}}(A-xI) = p_0 + p_1x + \dots + p_n x^n$....

**2**

votes

**0**answers

255 views

### Function recursion relation over symmetric group

Hi!
Let P be a permutation in the symmetric group SN and let π=πj, j+1 be a transposition of elements j and j+1 of the permutation. Let A(P) be a function in dependence of the permutation P. P&...

**1**

vote

**1**answer

542 views

### Hankel determinants of symmetric functions

The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions.
...

**23**

votes

**3**answers

2k views

### Splitting the determinant polynomial into linear factors - a Dedekind problem

Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial
$\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq i\...

**11**

votes

**2**answers

1k views

### What are picard categories, where can I learn more about them, and why should I care to?

I have the category-theoretic background of the occasional stroll through Mac Lane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor ...

**6**

votes

**3**answers

2k views

### Sarrus determinant rule: references, extensions

SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS
An undergraduate came to me with an identity for 4x4 determinants that is actually correct:
$\det(A)=h(A)+h(RA)+h(R^{2}A)$
where R cyclically ...

**18**

votes

**2**answers

1k views

### Lifting matrices mod 2 to integers.

The following question was motivated by my research.
Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs ...

**2**

votes

**2**answers

2k views

### Statement of Lagrange's theorem on determinants(elementary question).

Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...

**5**

votes

**2**answers

678 views

### What's the correct notion of determinant of a bilinear pairing?

By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...