# Tagged Questions

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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### 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 ...
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### Proving that $\det(A) = 0$ when the columns are linearly dependent

Proposition: Let $A$ be a $(n \times n)$-matrix. If the columns of $A$ are linearly dependent, then $\det(A) = 0$. Attempt at proof: Let $A = (A_1, A_2, \ldots, A_n)$, where each $A_i$ is a column ...
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### 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 $B$ ...
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### 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 ...
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### 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 ...
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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} \\-1&a_{2,2}&a_{2,3}&... 0answers 95 views ### The determinant of a special matrix Recently, I encounter the problem of calculating the determinant of the following matrix $$\left(\begin{array}{cccc} \sin(\theta_1) & \sin(\theta_1 + \delta_1) & \cdots & \sin(\theta_1 + (... 0answers 167 views ### Prove that \phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)]. I have proved these two exercises: (1) Suppose that T \in \Lambda^p(V^*) and v_1, \ldots, v_p \in V are linearly dependent. Prove that T(v_1, \ldots, v_p) = 0 for all T \in \Lambda^p(V^*). ... 0answers 437 views ### Understanding a proof about Hilbert Matrix EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO. Lately I've been interested in the Hilbert Matrix (its definition will come later). I went ... 0answers 21 views ### Determinant of \delta function Let$$\delta_i^j=\left\{ \begin{aligned} 1 ~~~~~~i=j \\ 0 ~~~~~~i\ne j \end{aligned} \right. $$1\le i,j\le n. How to prove$$ \begin{vmatrix} \delta_{j_1}^{i_1} ~...~ \delta_{j_n}^{i_1} \\ \\ \... 0answers 31 views ### Proof of result involving Pfaffian of a matrix Show that $$\text{Pf} MAM^T = \text{det}M \cdot \text{Pf} A$$ for any matrixM$and antisymmetric$A$. Attempt: $$\text{Pf} MAM^T = \frac{1}{2^N N!} \epsilon_{\alpha_1 \dots \alpha_{2N}} (MAM^T)_{\... 0answers 108 views ### Determinant of \vert A' C A\vert Let X be an n\times p real matrix with column rank k, where 0<k<p<n, and let A be a semi-orthogonal matrix (the columns are orthonormal) such that A'X=0, i.e. the column space of ... 0answers 104 views ### 2\times 2 block Toeplitz determinant My question is about computing asymptotic the determinant (dimension of the matrix n\to\infty) of a 2\times 2 block Toeplitz matrix.$$\mbox{det}\left(\begin{array}{cc} a_n & b_n \\ d_n & ... 0answers 64 views ### Calculate Determinant A size n I am given homework like this, calculate the Matrix $$\begin{bmatrix}x+1 &x&x&...&x\\x&x+2&x&...&x\\x&x&x+3&...&x\\...&...&...&...&...\... 0answers 129 views ### Determinant of non-square Jacobian Suppose I have a 3d solid in {\bf R}^4 which can be parametrized by the function F:W\subset{\bf R}^3\rightarrow{\bf R}^4. Now suppose I want to calculate the volume of this solid. Then naively I ... 0answers 69 views ### Help me to prove the determinant of given matrix. Suppose, M=\begin{bmatrix}\begin{array}{ccccccc} -x & a_2&a_3&a_4&\cdots &a_n\\ a_{1}+x & -x-a_2 & 0&0&\cdots &0\\ a_1+x&0 & -x-a_3 &... 0answers 28 views ### Jacobian Determinant vs. Divergence for local expansion I am interested in image processing (in 3D). I often see two different ways of measuring local expansion or contraction of a deformation: the Jacobian determinant or the divergence (but usually the ... 0answers 62 views ### Is there a name for this generalization of the determinant? In the context of averaging over network paths, I arrived at a certain generalization of the determinant for an n\times n square matrix A, that is$$D_k(A) := \sum_{(j_1,j_2,...,j_n):\,\, |\{j_1,.... 0answers 21 views ### generalizations of the linearly independence of column vectors in a Vandemonde matrix to higher dimensions Let$x_1,x_2,\cdots,x_n\in\mathbb{R} $or$\mathbb{C}$. By the non-degeneracy of Vandemonde matrix the maps $$f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$$$x\longmapsto (1,x,x^2,\cdots,x^{n-1}... 0answers 71 views ### How does the determinant change with respect to a base change? Problem Suppose$k$is a (commutative) field, and$A$is a finite (dimensional) commutative unitary$k$-algebra.$M=A^n$is a free$A$-module, and therefore can be seen as a finite-dimensional$k$-... 0answers 55 views ### Can the determinant of an integer matrix with$k$given rows be the gcd of the determinants of the$k\times k$minors of those rows? I'm interested if the following is true: Let$n\geq k\geq1$be integers, let$A\in\mathbb Z^{k\times n}$and denote the$\binom nkk\times k$minors of$A$by$A_1,\ldots,A_N$. Then the ... 0answers 34 views ### Over what rings is the Hefferonian determinant unique? Fix an$n\in\mathbb{N}$and a field$\mathbb{K}$. A lot of texts in linear algebra like to define the determinant function on$\operatorname{M}_n\left(\mathbb{K}\right)$as the unique function$\...
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 ...