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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Finding the determinant of a $3*3$ matrix that was developed using LaGrangian multiplier to optimize the surface area of a conical frustum
This is the matrix I wrote down.
I'm struggling to find its determinant as by hand it would be impractical and would take a lot of time, is there any online calculator or a way to find it quickly, if ...
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Identity about determinant of $I-A$ [duplicate]
I was reading the book Introduction to Random Matrices by Anderson et al and I think that the authors are using the following characterization of $\det(I-A)$ in Lemma $3.2.4$ in the book.
$$\det(I-A)=...
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Prove this conjecture: Two lists of vectors are in the same orientation iff this transformation exists
Can you please help prove or disprove my conjecture below?
Definition: Given vector space $V$ of $n$ dimensions, and ordered lists $x, y$ such that $x,y \subset V$; $x,y$ are each linearly independent;...
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Proof that the characteristic polynomial of a matrix can be expressed as a sum of the matrix's determinant and scaled powers of its eigen values
In the book, Introduction to Computational Linear Algebra, the authors state the following:
For any matrix $A \in \mathbb{R}^n$, its characteristic polynomial, $p_\text{A}(\lambda) = \text{det}(A - \...
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Determinant of a tridiagonal matrix with constant values $5,3,2$ on the main, upper and lower diagonals
Can you help me to compute determinant of matrix
$$A =\begin{pmatrix}
5 &3 &0 &\cdots &0 &0 \\
2&5 &3 &\cdots &0 &0 \\
0&2 &5 &\...
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+50
On the height of the Jacobian ideal of the determinant of a square matrix of variables
Let $k$ be a field of characteristic $0$, let $\mathbf X=[X_{ij}]_{1\le i,j\le n} $ be a square matrix of indeterminates where $n\ge 2$. Consider the polynomial $f(\mathbf X)=\text{det}(\mathbf X)\in ...
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Determinant of involving adjacency matrix of subgraph
Let $A$ be the adjacency matrix of a graph $G$ labelled $\{x_1,\cdots,x_n\}$. Let $B:=(I-\frac{1}{2d}A)^{-1}$ where $d$ is a positive integer.
Then let $A'$ be the adjacency matrix of the graph $G\...
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How many different determinants are possible by reordering entries?
I was preparing a worksheet about cofactor expansion of $3\times3$ determinants and decided to create only matrices of order 3 whose entries are among $\pm1,\pm2,\dots,\pm9$ such that the absolute ...
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Does $\det(AB) = \det(A) \det(B)$ hold for differential operators $A$ and $B$?
Following the inconsistency I was facing in a previous post, I was wondering if the following property
$$ \det(A) \det(B) = \det(AB) $$
holds for generic $A$ and $B$ differential operators. It looks ...
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Proving a statement on linear algebra involving inequalities, determinants, and eigenvalues.
Suppose $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$.
I have observed by considering many examples of $x,...
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polynomial representation of determinant
The question I'm working on is, given a matrix $A = (a_{i,j})_{i,j = 1}^n$ and the polynomial
\begin{gather*}
P(x) : = \det\begin{bmatrix}a_{1,1} + x & a_{1,2} + x & ... & a_{1,n} + x\\...
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Are alternate theories of determinants equivalent?
Let $d: \mathbb F^{m \times n} \to \mathbb F$ be a function from real matrices to a scalar which is row linear, meaning:
(i) For any matrix $M,$ if $M'$ is $M$ with a single row multiplied by ...
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Determinant of a polynomial: Exercise from chapter 1, "Linear Algebra" by Georgi E. Shilov
I am solving the exercise from "Linear Algebra" by Georgi E. Shilov. The question asks to prove the following:
\begin{align*}
\begin{vmatrix}
1 & 1 & \cdots & 1 \\
x_1 & x_2 ...
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Singular matrix, non-vanishing principal minors
Is there a nice way to characterize the singular matrices whose principal minors up to some size $r\times r$ are always non-vanishing?
I was wondering whether there is some theorem regarding this, ...
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Functional determinant inconsistency
While trying to compute functional determinants, I faced an inconsistency, which I can exemplify by defining the following matrix
$$ M = \begin{pmatrix} i \frac{d}{dt} + t & 0 \\\ 0 & i \frac{...
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Symmetry in factor theorem in Determinants
Once my teacher had told me during driving the value of a standard determinant that confused me till now.
The value exactly was (a-b)(b-c)(c-a)(a+b+c).
Here,
(a-b)(b-c)(c-a) had come from factor ...
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Sums of vector triangles and a resultant identity of determinants
In the $\mathbb R^2$ plane, let $a, b, c$ be vectors, let $o$ be the origin, and let $d = b + c$. Determine the area of $\triangle oad$ in terms of $\triangle oab$ and $\triangle oac$, and infer an ...
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When applying Crout's method for LU decomposition, when will the determinant be negative?
I'm attempting to write a software library for handling matrix calculations. The implementation for calculating the determinant I've written consists of decomposing a matrix $A$ into a lower ...
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Fast determinant of an Alexander matrix
I want to compute the determinant of a polynomial $n \times n$ matrix where each entry is a univariate polynomial of degree at most $1$. I calculated it naively and was quickly reminded of how fast $n!...
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Compute the determinant of the following Hankel matrix involving factorials: a$_{ij}=(i+j-1)!$
Let A be an $n\times n$ square matrix with a$_{ij} = (i+j-1)!$. Compute $\text{det}(A)$.
I've tried to factorise from each column the elements of the first row and then performed some row operations ...
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Relationship of determinants of block matrices
I am thinking about the following problem:
Let $$M = \begin{bmatrix}A & X_{12} & X_{13} \\ X_{21} & B & X_{23} \\ X_{31} & X_{32} & C\end{bmatrix} \in \mathbb{R}_{\geq0}^{n \...
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Is a real-symmetric matrix diagonalizable by a special-orthogonal matrix? [duplicate]
Suppose I have a real-symmetric matrix $X$. The spectral theorem guarantees that $X = O^T D O$ for some orthogonal matrix $O$ and diagonal matrix $D$. However, I would like to be able to diagonalize $...
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Solving a matrix equation involving trace and determinant
Let $n\in\mathbb{Z}_{>1}$. Consider the following set :
$$
E_n=\{A\in\mathcal{M}_n(\mathbb{C})| \mathrm{Tr}(A)=\det(A)\}
$$
My goal is to characterize $E_n$ for any $n$.
I have already found a ...
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Expectation of log-determinant of sample covariance
Suppose $x_1,\dots, x_n\sim \mu$ are iid samples in $R^d$ with mean zero and identity covariance: $E_{x\sim \mu} x=0, E_{x\sim \mu} x x^\top =I.$ Let $C_n$ denote sample covariance $C_n :=\frac1n x_i ...
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Show that $det(A)$ is divisible with the sum of all elements in $A$.
We have $A(3×3)$ matrix (the sum of the elements on each row, column and diagonal are the same) with non-zero natural entries. Show that $det(A)$
is divisible with the sum of all elements in $A$.
I ...
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Determinant of order 2 block matrix following specific instructions
Given a block matrix of the form
$$A = \begin{pmatrix}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{pmatrix}$$
with $A_{11}$ invertible, I want to prove that $$\det (A) = \det (A_{11}) \det \left(...
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Determinant of Green function
Let $f : \Bbb Z^d \to \Bbb R$. The discrete average value $\overline{f(x)} : \Bbb Z^d \to \Bbb R$ is defined by $$\overline{f(x)}:=\frac{1}{2d}\sum_{y \in S_x}f(y)$$ where $S_x$ are the neighbors of $...
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Determinant of a continuous Kernel
Before jumping to the determinant, lets consider the discrete eigenvalue equation:
$$A\text{x}=\lambda\text{x}$$
$$\sum_jA_{ij}\text{x}_j=\lambda\text{x}_i$$
And the continuous eigenvalue equation:
$$\...
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Relationship between $A$ and $\frac{1}{2}(A + A^T)$?
Is there a general Relationship between the determinant (or eigenvalues/-vectors) of a square matrix $A$ and its "symmetrification" $\frac{1}{2}(A + A^T)$?
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determinant of multiple Matrices -question from an old exam-
translation from Hebrew:
let $A=(a_{ij})_{n\cdot n}$ prove that the determinants of the following matrices is as follows: $(u_1+u_2+...+u_n)det(A)$.
I proposed using the additive property of the ...
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Binet's Formula Proof (Einstein notation, Levi-Civita symbol)
I'm trying to understand the proof below... Although I'm familiar with the notation itself, I'm in trouble with the underlined statement... can anyone explain?
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How to calculate the determinant of this matrix (Hessian matrix + Identity)
I am trying that in order to calculate the volume under a graph of a function $\phi : \mathbb{R}^k \to \mathbb{R}^{k+1}$ the Area element we should integrate is $$ \sqrt{1+\left|\nabla\phi\left(x\...
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Alternative (intuitive) proof of change of variables for integration(Jacobian)
When proving the Jacobian matrix for change of variables, you are usually given the linear transformation approach, computing that "transformed" area in the x,y plane using the determinant, ...
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Prove that $\det\begin{pmatrix}A&B\\-B&A\end{pmatrix}$ is a sum of squares of polynomials
As discussed for example in this question, given any pair of real squared matrices $A,B$ we have the identity
$$|\det(A+iB)|^2 = \det\begin{pmatrix}A&B\\ -B&A\end{pmatrix}.$$
In particular, ...
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A probabilistic proof of Oppenheim's inequality?
Oppenheim's inequality is a standard result about the Hadamard product of positive definite matrices. It goes as follows, let $A=(a_{ij})_{i,j\leq n},B=(b_{ij})_{i,j\leq n} \in S_n^{++}$ where $S_n^{++...
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Is there a less time-consuming way to solve a Symmetric Matrix Equation
I'm currently working on solving an equation that involves a symmetric matrix C with 4 unknown variables, and a vector A of the same dimension. The equation I'm trying to solve is:
...
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find the characteristic polynomial of $I_n +xy^*$. [duplicate]
let $x,y \in \mathbb C^n$ find the characteristic polynomial of $I_n +xy^*$.
I think we can use the relation $X^nP_{AB}(x)=X^mP_{BA}(X)$
we put $A=x$ and $B=x^{-1}+y^*$ is this right?
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Confusion regarding wedge products
While studying smooth manifolds and differential forms I have come across multiple definitions of the wedge product, and I have been having some trouble seeing the equivalence between them.
At the ...
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Best (or correct) approach on finding characteristic polynomial of this $4 \times 4$ matrix
Let $f: \mathbb{R}^4 \to \mathbb{R}^4$ be a linear transformation with matrix representation
\begin{align*}
A = \begin{pmatrix}
-5 & -5 & -9 & 7 \\
8 & 9 & 18 &...
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Can anybody provide a proof for the general Hadamard determinant theorem?
For a general matrix $B=\left[b_{i j}\right] \in \mathbb{C}^{n \times n}$
$$
|\operatorname{det} B| \leq \prod_{i=1}^n\left(\sum_{j=1}^n\left|b_{i j}\right|^2\right)^{1 / 2} \text { and }|\...
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Show that $XY=0$ or $YX=0$
We have $X,Y$ $(2×2)$ matrices with complex entries and $X=A^{2}-B^{2}$ and $Y=AB-BA$. We know that $\det(X)=\det(Y)=0$. Show that $XY=0$ or $YX=0$.
I see that Trace of $Y$ is $0$ and $\det(Y)$ is ...
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General solution of a system with distinct eigenvalues
I'm going through Landau's Mechanics, and on the chapter about oscillations of systems with more than one degree of freedom, he goes from the characteristic equation
$$\sum_{k}(-\omega^2m_{ik}+k_{ik})...
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the relation between $p_{AB}$ and $p_{BA}$ such that $p$ is characteristc polynomial.
Let $A\in M_{m,n}(\mathbb C)$ and $B\in M_{n,m}(\mathbb C)$ s.t $m\leq n$
Calculate the product $$\begin{pmatrix}
I_m & -A \\
0_{n,m} & I_n \\
\end{pmatrix}\begin{pmatrix}
AB & O_{m,n} \\
...
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Matrix manipulations with Levi-Civita symbol
My question relates to this reply on math.stackexchange.
More precisely, I am wondering about the following sequence of expressions involving elements of an invertible square matrix $M$ and a pair of ...
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Determinant of Integral operators
According to Wikipedia, the Fredholm determinant of $1-T$ with a trace-class integral operator $T$ with kernel $K$ is ("informally") defined by
\begin{equation}
\text{det}_F\big(1-T\big)=\...
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Find matrix $x, y, z$ from its eigenvalues.
If I were given a matrix $A$,
$$ A =
\begin{bmatrix}
5 & -2 & 3 \\
0 & y & 0 \\
x & 7 & z \\
\end{bmatrix}
$$
with corresponding eigenvalues $1, 7,$ and $−4$, respectively, ...
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determinant as product of eigenvalues
I was reviewing the Wikipedia page for determinants which states that for all complex matrices, the determinant is the product of its eigenvalues. However, isn't this true for real matrices as well, ...
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Problem about matrices and determinants
I have a problem about matrices, but the problem is that I don't know enough about them to know the answer (and I am not a native speaker, so I'll try my best to describe the problem).
So, imagine ...
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Proof of the determinant of the Vandermonde matrix via induction
Let $V_n$ be the Vandermonde matrix made up of a sequence of $n$ real numbers $\lambda_1, ..., \lambda_n$,
\begin{align*}
\begin{pmatrix}
1 & \ldots & 1 \\
\lambda_1 & ...
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Determinant of matrix with main diagonal of natural numbers [duplicate]
Determine the $\det(A)$ by induction, where:
$$A = \begin{pmatrix}
1 & x & x & . & . & . & x \\
x & 2 & x & . & . & . & x \\
x & x & 3 & . ...
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