# Questions tagged [determinant]

Questions about determinants: their computation or their 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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### Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
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### 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 ...
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### 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 ...
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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 $\... 0answers 58 views ### Rank of a matrix whose all entries have the form$m^k$The original problem is: Compute the determinant $$\begin{vmatrix} 1^k & 2^k & 3^k & \cdots & n^k \\ 2^k& 3^k & 4^k &\cdots & (n+1)^k \\ 3^k& 4^k &... 1answer 276 views ### A challenge question in determinant of real matrices! Suppose that n\in \mathbb N -\{1\} and a_{11},a_{12},\ldots,a_{nn} are n^2 distinct real numbers, prove that there is some enumeration of a_{ij}'s like b_{ij}\ (i,j=1,2,\ldots,n) such that,... 0answers 264 views ### Determinant expression for the power sum Let S_{n,r} := \sum_{k=1}^{n} k^r be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression:$$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det \begin{... 0answers 87 views ### Show$\mathrm{det}(M)$is well defined. One intuitive way to approach studying the determinant of a given matrix$M$is to inspire its formal definition in the signed volume of applying the corresponding transformation to the unit$n$-cube. ... 0answers 61 views ### Maximal determinants of zero-one matrices Is it possible to find the maximal determinant of matrices in$\{0,1\}^{n \times n}$? If so, what do matrices with maximal determinant look like? For example, when$n=3$, it's not hard to see that ... 0answers 66 views ### Abtract explanation for$\det(A+B) = ∑_{k=0}^n \langle \Lambda^k A, \Lambda^{n-k} B \rangle$Let$A$and$B$be$n×n$matrices. If we expand the determinant of$A+B$as a sum over all permutations of$[\![1,n]\!]$and all choices of whether the coefficient comes from$A$or from$B$, this ... 0answers 33 views ### Matrices with positive permutation products Let$A=(a_{ij})$be a$n\times n$real matrix such that $$\operatorname{sign}(\sigma) \cdot a_{1,\sigma(1)} a_{2,\sigma(2)} \ldots a_{n,\sigma(n)}\ge 0$$ for all$\sigma\in S_n$. Is there a name ... 0answers 79 views ### Methods of solving roots involving matrix determinant I have a square matrix$F$whose elements depend non-linearly on a complex parameter$s$. I would like to know the values of$s$such that$\det(F)=0$, i.e., those$s$that make$F$singular. I would ... 0answers 107 views ### Characterisation of matrices whose real eigenvalues are positive My question is the following Is there a characterisation of$n\times n$matrices with real entries whose real eigenvalues are positive? I am interested in this question because I am analysing some ... 1answer 123 views ### Finding the minimal value of a$4\times 4$determinant The question. Let$\xi=(\xi_1,\xi_2,\xi_3,\xi_4)\in\mathbb R^4$be a vector with irrational coordinates. I am interested in finding the minimal value$\mu_\xiof \left\vert \det \begin{pmatrix} ... 0answers 93 views ### Prove that the matrix [\Gamma(\lambda_{i}+\mu_{j})] is nonsingular. Let A be an n\times n matrix whose entries are \begin{align*} a_{ij} = [\Gamma(\lambda_{i}+\mu_{j})] \end{align*} where 0 < \lambda_{1} < \ldots < \lambda_{n} and 0 < \mu_{1} < \... 0answers 67 views ### The polarization of the determinant is invariant? Given n \in \mathbb N, I am asked to show that there is a multilinear symmetric \operatorname{GL}_n-invariant form \phi : (M_{n \times n})^l \to \mathbb R (for some l \geq 0) such that \phi(A,... 0answers 39 views ### Sufficient/necessary condition for submatrix determinant (minor) that decreases with size? Definition. Given a square matrix {\bf{A}}=[a_{ij}] \in {\mathbb{C}^{n \times n}}, the submatrix {{\bf{A}}_{{i_1},{i_2},...{i_k}}} is formed by retaining the ({i_1},{i_2},...{i_k})-th rows and ... 0answers 70 views ### Find the values of a \in \mathbb{R} where the system Ax=x allows a solution different to the null one I have to find the values of a \in \mathbb{R} where the system Ax=x allows a solution different to the null one, and then solve the system with those values I found of the following matrix: ... 0answers 276 views ### Calculation of the Pfaffian of a matrix I have a set of N numbers \lbrace \lambda_i\rbrace_{i\in[1,N]} that belong to [0,2\pi[ and a real number L and I am trying to evaluate the following Pfaffian expression.\mathrm{Pf}\left(\... 0answers 112 views ### How many subsets ofn$linearly independent binary strings of length$n$? Let's consider binary words of length$n$with elements {-1,1}. There are$2^n$binary words of length$n$. Now let's consider a subset of$n$such binary words. All possible subsets are$\binom{2^n}{...
I am trying to understand the determinantal approach on Harris book "Algebraic Geometry: A first course" on proving that the intersection of two quadrics containing the twisted cubic in $\mathbb{P}^3$ ...