## New answers tagged determinants

5
votes

Accepted

### Results of invertibility of a matrix involving the Szego kernel

Expand the determinant $D=\det k(x_j,y_k)$ along the first row. This shows that as a function of $x=x_1$, it is of the form
$$
D(x) = \sum_{j=1}^n \frac{c_j}{1-y_j x} ,
$$
with $c_j$ independent of $x=...

0
votes

### Results of invertibility of a matrix involving the Szego kernel

Sorry for my previous answer. This is a partial answer for the $2\times 2$ case.
Notice first that we can assume without loss of generality that $z_1=0$.
Otherwise we can apply the Moebius ...

0
votes

### Determinant of Jacobian and directional derivatives

As you point out, there are some general matrix facts in here. Consider the eigendecomposition of a matrix: $A = S \Lambda S^{-1}$, where $S$ and $S^{-1}$ are orthonormal (implying their determinant ...

6
votes

### Maximum determinant of binary matrices with special properties

I'll give a partial answer to your Question #2. If you know $k$ and also how many entries are equal to $1$ in each column, you can actually compute the absolute value of the determinant exactly.
By ...

2
votes

### Maximum determinant of binary matrices with special properties

In case 1) a cheap bound is $|{\rm det}(A)|\le k^n$. That's because $A/k$ is a stochastic matrix and therefore has all its eigenvalues in the closed unit circle. See https://en.wikipedia.org/wiki/...

2
votes

### Testing for equal characteristic polynomials using a single determinant calculation

The coefficients of the characteristic polynomial are bounded by $n!$ so the coefficients of the difference are bounded by $2 n!$. The leading coefficient of the difference is integer, so its absolute ...

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