# Tag Info

Accepted

• 83.2k
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
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 ...

### 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
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
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

### 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
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
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
Accepted

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

### 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