Question about determinants, computation or theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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17
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245 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
14
votes
0answers
507 views

Prove this determinant identity combinatorially

This is for those of you who understand the Lindstrom-Gessel-Viennot lemma. I am looking for a proof of the following identity using paths and such: Let $A$ be an $n\times n$ matrix, and for ...
7
votes
0answers
217 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix. I have already asked a (viewed but unanswered) ...
7
votes
0answers
223 views

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
7
votes
0answers
110 views

Maximum determinant of latin squares

I strongly conjecture that the maximum absolute determinant of a latin square can be attained by a circulant matrix. For example, $\pmatrix {5&4&2&3&1 \\ 1&5&4&2&3 \\ ...
6
votes
0answers
37 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But ...
6
votes
0answers
229 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha ...
6
votes
0answers
413 views

Determinant of Transpose of Linear Map

I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's ...
5
votes
0answers
93 views

calculation of the determinant of a block matrix little help

I need to prove $$\operatorname{det}\begin{pmatrix}A & B \\ C & D\\ \end{pmatrix}= \operatorname{det}(DA-CB),$$ where $A,B,C,D \in M_{n\times n}(R)$ with the property that $A$ and $B$ ...
5
votes
0answers
126 views

How to prove the determinant?

We have to prove the following result without expanding $\left|\begin{array}{lll} a^3 & a^2 &1 \\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} ...
5
votes
0answers
576 views

Linearity of the determinant

I'd like to prove the following properties of the determinant map. $\det I = 1$ $\det$ is linear in the rows of the input matrix The determinant map is defined on $n\times n$ matrices $A$ by: ...
5
votes
0answers
365 views

Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.

(this is an improved version of What about other symmetric functions of the eigenvalues? ) Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
147 views

Determinant expression for the power sum

Let $S_{n,r} := \sum_{k=1}^{n} k^r$ be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression: $$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det ...
4
votes
0answers
56 views

How can I determine the sign of a term in a determinant when the indices are out of order?

I am reading Shilov's book linear algebra. He explains how to compute determinants. Basically, for the plus terms you write \begin{equation} x_{a1}x_{b2}x_{c3}x_{d4}x_{e5} x_{f6} \end{equation} and ...
4
votes
0answers
46 views

Rank of a matrix whose all entries have the form $m^k$

The original problem is: Compute the determinant $$\begin{vmatrix} 1^k & 2^k & 3^k & \cdots & n^k \\ 2^k& 3^k & 4^k &\cdots & (n+1)^k \\ 3^k& 4^k ...
4
votes
0answers
76 views

Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
4
votes
0answers
87 views

Expectation of the absolut value of the determinant of a random matrix

Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable $$X := |\det A|$$ Can ...
4
votes
0answers
44 views

Determinant of triangular matrix except for one column (atomic/Gauss/Frobenius)

Is there some "smart" way to calculate determinants that look like this? $\begin{vmatrix}-1&a_{1,2}&a_{1,3}&a_{1,4}&\cdots&a_{1,m-1}&a_{1,m} ...
4
votes
0answers
89 views

The determinant of a special matrix

Recently, I encounter the problem of calculating the determinant of the following matrix $$\left(\begin{array}{cccc} \sin(\theta_1) & \sin(\theta_1 + \delta_1) & \cdots & \sin(\theta_1 + ...
4
votes
0answers
368 views

Understanding a proof about Hilbert Matrix

EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO. Lately I've been interested in the Hilbert Matrix (its definition will come later). I went ...
3
votes
0answers
66 views

Proving generalized Cassini's identity using determinant?

Motivation It is not hard to show, by using the general solution, that Proposition. If $(a_{n})_{n\in\Bbb{Z}}$ satisfies the recursive formula $ a_{n+2} = pa_{n+1} + qa_{n}$, then for any $n, i, ...
3
votes
0answers
41 views

Distinct $0,1$ symmetric circulant determinants

If $M$ is a circulant integer matrix of size $n\times n$ whose entries are randomly chosen from $\{0,1\}$ value, how many different determinants does $M$ possibly take value in? If $M$ is symmetric ...
3
votes
0answers
51 views

Finding $n$ scalars such that $\det{(cI-A)}=0$ without eigenvalues

My problem is this Let $A$ be an $n\times n$ matrix over $\mathbb{F}$. Prove there are at most $n$ distinct scalars $c\in\mathbb{F}$ such that $\det{(cI-A)}=0.$ I know that the determinant is ...
3
votes
0answers
65 views

Prove that the determinant of a given matrix is proportional to the area of the triangle whose corners are the three points.

For three points in 2D, $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, show that the determinant of \begin{bmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 2\\ x_3 & y_3 & 3\\ ...
3
votes
0answers
72 views

Probability that a random integer matrix is singular

Let $A$ be a $n\times n$-matrix with integers in the range $u..v$ , where $u<v$ are arbitary integers. Is there a formula, or at least, a good estimate, for the probability that the matrix is ...
3
votes
0answers
42 views

Determinant of a generalization of Moore matrices

The Moore matrix over $\mathbb{F}_q$ is the $n\times n$ matrix whose i'th row is: $a_i,a_i^q,a_i^{q^2},\dots,a_i^{q^{n-1}}$. The determinant of this matrix is the product of all linear combinations ...
3
votes
0answers
77 views

Invariants under a transformation

Consider a $j=1,\,SU(2)$ representation (or fundamental $SO(3)$ representation). Suppose that $a_1, b_i, c_i$ with $i=1,2,3$ are vectors transforming under this representation i.e ...
3
votes
0answers
52 views

How to prove that the determinant is the same no matter how you take it?

To find the determinant, pick a row and move along it creating minors and use the recursive definition of determinant. How do we know that the determinant will be the same no matter which row you ...
3
votes
0answers
173 views

Determinant proof

Let $A\in M_n(\mathbb C)$ and $\alpha \in \mathbb C$. If $B$ is the matrix obtained by multiplying a single row of $A$ by $\alpha$, then det$(B)=$ $\alpha$ det$(A)$. I'm trying to understand and use ...
3
votes
0answers
95 views

Prove that $\phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)].$

I have proved these two exercises: (1) Suppose that $T \in \Lambda^p(V^*)$ and $v_1, \ldots, v_p \in V$ are linearly dependent. Prove that $T(v_1, \ldots, v_p) = 0$ for all $T \in \Lambda^p(V^*)$. ...
3
votes
0answers
155 views

How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?

The question is pretty straight-forward: how do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$ ?
3
votes
0answers
79 views

Identifying factors of higher order in a determinant

Consider a $n\times n$ matrix $A$ whose elements are some polynomials in the indeterminates $x_1, x_2,\ldots,x_m$. To calculate the determinant of such a matrix, one of the usual ways is to treat the ...
2
votes
0answers
57 views

Help me to prove the determinant formula

Actually it is about the question of n-linear function, but it is so relevant to the determinant formula. Here is the notation of the theorem. If $n>1$ and $A$ is an $n \times n$ matrix over $K$, ...
2
votes
0answers
17 views

Proof of Leibniz formula from Laplace expansion

I'm trying to prove Leibniz formula for the determinant using Laplace expansion. Here's my attempt: For a $1 \times 1$ matrix $A = \begin{pmatrix}a_{11}\end{pmatrix}$, define $\det A = a_{11}$. For ...
2
votes
0answers
56 views

The Maximum Singular Value of a Certain $\{1,-1\}$-Matrix

I have a $16$-dimensional real symmetric matrix with entries in $\{1,-1\}$. $11$ of the rows are pairwise orthogonal, so are the remaining $5$ rows. But the two orthogonal sets are not necessarily ...
2
votes
0answers
47 views

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2^{n−1}$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$.

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2^{n−1}$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$. How to prove this result? (I found this statement while ...
2
votes
0answers
52 views

Can I perform elementary line operations to $\det(A-\lambda I)$ like this?

I have a $4 \times 4$ matrix: $$A = \begin{pmatrix} 2 & 3 & 1 & 0 \\ 4 & -2 & 0 & -3\\ 8 & -1 & 2 & 1\\ 1 & 0 & 3 ...
2
votes
0answers
31 views

Show $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$

$A$ is a invertible $n \times n$ matrix. Show: $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$ Can someone give me a hint on how to show that?
2
votes
0answers
31 views

Geometric Interpretation of Determinant of Transpose

Below are two well-known statements regarding the determinant function: When $A$ is a square matrix, $\det(A)$ is the signed volume of the parallelepiped whose edges are columns of $A$. When $A$ is ...
2
votes
0answers
48 views

Linear Algebra - Determinant Properties

A = \begin{bmatrix} a & b & c \\[0.3em] d & e & f \\[0.3em] g & h & i \end{bmatrix} B = \begin{bmatrix} g & ...
2
votes
0answers
21 views

why is the sum over even and odd permutations the same?

let $m$ be an $n \times n$ matrix (over $\mathbb{R}$,say) and for a permutation $\sigma \in S_n$ define the monomial: $$ P_\sigma(M) = \prod_{j=1}^n m_{j,\sigma(j)} $$ let $\tau$ be an odd ...
2
votes
0answers
76 views

How to prove the following identity?

For any even integer $N$, define two sets $$K_+=\left\{\frac{(2m+1)\pi}{N}|m=-\frac{N}{2},-\frac{N}{2}+1,...,\frac{N}{2}-1\right\}$$ and ...
2
votes
0answers
44 views

How to compute determinant (or eigenvalues) of this matrix?

Let us have the $n \times n$ circulant matrix given by \begin{equation} C(c_0,c_1,\cdots, c_{n-1}) =\begin{bmatrix} c_0 & c_1 & c_2 &\cdots & c_{n-1}\\ c_{n-1} & c_0 & c_1 ...
2
votes
0answers
44 views

Value of determinant using given conditions.

Let $A$ be a $2$ x $2$ matrix with real entries and $det(A)$ is equal to $d$ which is non-zero. It is given that $det(A +d(adjA))=0$ where $adj$ stands for the adjoint of the matrix. We have to find ...
2
votes
0answers
23 views

How to compute the following Jacobian

I need to show that the Jacobian of the n-dimensional spherical coordinates is $$\displaystyle r^{n-1}\sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots\sin\phi_{n-2}$$ then I have computed the Jacobian matrix, ...
2
votes
0answers
31 views

Determinants, Pfaffians, and…?

I recently stumbled across the wikipedia entry on Pfaffians and found them rather interesting, especially the property below. (assuming $A$ is a $2n\times 2n$ skew symmetric matrix) ...
2
votes
0answers
62 views

Is the determinant an analytic function?

I came accross a paper stating that the analytical property of determinants of complex matrices allows us to use some theorem for analytic functions. I am not able to confirm this since I am not sure ...
2
votes
0answers
56 views

Hypermatrices, hyperdeterminants and Grassmannians.

Let $Gr(k,n)$ the Grassmannian manifold of the $k$-planes in $\mathbb{C}^n$ and consider the Plucker embedding $\pi: Gr(k,n) \to \mathbb{P}(\Lambda^k \mathbb{C}^n)$. Let $A$ be the set of $n \times n$ ...
2
votes
0answers
44 views

Is the Cone over Grassmannian manifold a determinantal variety?

Let consider the Grassmann manifold $Gr(k,n)$ in the Plucker embedding and the Cone over $Gr(k,n)$, say $C(Gr(k,n))$. On the other hand consider $M$ the set of $n \times n$ skew-symmetric matrices. ...
2
votes
0answers
41 views

Log concavity/convexity of a determinant

I was wondering if anyone would be able to help me determine whether the following quantity is log concave or not with respect to $\alpha$? $$\left[\det(\textbf Y^\top \textbf P \textbf G \textbf ...