Questions tagged [determinants]
Questions about the determinant of square matrices or linear endomorphisms. Also for closely related topics such as minors or regularized determinants.
500
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0
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range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants
The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
- 723
4
votes
1
answer
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Reconstructing a (unitary) matrix from the determinant of its sub-matrices
I want to find the unitary $N \times N$ matrix U from the following data. Let $M$ be an integer $(1< M<N-1)$ and let $\mathcal S$ be the space of all the possible subsets of $\{1,2,\dots, N\}$ ...
- 61
13
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2
answers
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Determinant of $V^* V$ where $V$ is rectangular Vandermonde matrix with nodes on unit circle
Let $z_{1},\dots,z_{k}$
be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$
consider the rectangular Vandermonde matrix
$$
V_{N}=\begin{pmatrix}1 &...
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0
votes
1
answer
445
views
A determinant problem with symmetric PSD matrices
Suppose we have a a set of matrices in the complex field of the form $a_iv_iv_i^H$ for $i=\{1,\dots,n\}$ where $a_i$ are constant positive real scalars and $v_i$ are constant complex valued finite ...
- 213
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votes
1
answer
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How to obtain the determinant of the difference of two matrices? [closed]
I am trying to obtain the determinant of the difference between the identity matrix and an A matrix. The question is such:
...
11
votes
2
answers
1k
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A binomial determinant fomula
Is there an existing or elementary proof of the determinant identity
$
\det_{1\le i,j\le n}\left( \binom{i}{2j}+ \binom{-i}{2j}\right)=1
$?
- 171
2
votes
3
answers
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LU decomposition
Consider a $N \times N$ symmetric real matrix $A$: $A_{ij} = (\sum_{k=1}^N n_{ik}) \delta_{ij} - n_{ij}$, where $n_{ij}$ is a real symmetric matrix whose elements are equal to $1$ or $0$. $A$ has one ...
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votes
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Has this generalization of a determinant (assigning multiplicities to the rows) been studied?
I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant:
Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
- 1,469
1
vote
1
answer
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views
Estimate the determinant of sparse 0-1 matrix
There is a matrix A where each entry is either 0 or 1. Each column has exactly a 1's and each row has at most b 1's. What's the upper bound of abs(|A|)?
The condition is stronger than the Hadamard's ...
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Given a correlation matrix $B$. What correlation matrix A (maximizes / minimizes) the following: det(A+B)
Given correlation matrix $B$ (positive semi-definite with ones in the diagonal).
1)Find the correlation matrix $A$ which maximizes $\det\left(A+B\right)$.
2)Find the correlation matrix $A$ which ...
- 41
2
votes
3
answers
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On matrices in linear forms with vanishing determinant
This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...
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2
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Is this a metric on the Grassmannian Manifold?
Let $m>n$ and consider the Set
$$S_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I_n \}.$$
Does the function $d\colon S_{m,n} \times S_{m,n} \rightarrow \mathbb{R}$ defined by
$$d(A,B)=\sqrt{1-\...
- 2,286
4
votes
1
answer
794
views
Determinant inequality of square-product sum of diagonal matrix and upper-triangular matrix
Recently, I have seen a matrix inequality but don't know how to prove it. The inequality goes as follows.
For an arbitrary $n\times n$ diagonal matrix $\mathbf{D}$ and an arbitrary upper-triangular ...
- 51
28
votes
2
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Determinants in Graph Theory
In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various linear-algebraic properties. For example, their trace can be calculated (...
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2
answers
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Determinant of non-symmetric sum of matrices
Given three real, symmetric matrices $A\succ0$ and $B$, $C⪰ 0$.
How can it be shown that:
$$\det(A^2+AB+AC) \leq \det(A^2 +BA +AC+BC) ? \qquad (\star)$$
Where $A^2$ is symmetric and positive ...
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10
votes
3
answers
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Derivative of a determinant of a matrix field
Let $A(x_1,...,x_n)$ be an $n\times n$ matrix field over $R^n$.
I am interested in the partial derivative determinant of $A$ in respect to $x_i$. In can be shown that:
$\frac{\partial{\det(A)}}{\...
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0
votes
1
answer
182
views
Find a generalized hypergeometric-based function yielding certain ratios of fifth-degree polynomials
Find a (presumably, generalized hypergeometric-based function $f(n,a,k)$), yielding for $n=1, a=\frac{1}{2}$,the rational function (ratio of fifth-degree polynomials)
\begin{equation}
f(1,\frac{1}{2},...
- 11
7
votes
1
answer
544
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Does this Linear Algebra Construction have a Name?
Let $\mathcal{R}$ be a ring and let $v^0,\ldots,v^{k-1}\in\mathcal{R}^m$ with $m \geq k$. Suppose we wish to find $w\in Span(v^0,\ldots,v^{k-1})$ such that $k-1$ specified coordinates of $w$ vanish (...
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votes
1
answer
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M-matrix plus S-matrix is P-matrix?
I am trying to prove that a mapping has a unique fixed-point by showing that its Jacobian is a P-matrix. In this particular case the Jacobian can be decomposed as the sum of two matrices and I would ...
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1
vote
1
answer
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Derivative of log determinant and inverse
I have a matrix $\Sigma$ with element $(i,j)$
$$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$
The matrix is positive definite and symmetric (it is a covariance matrix).
Now I need to evaluate
$$\frac{\...
- 29
0
votes
1
answer
622
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Gel'fand Yaglom functional determinant of non-diagonal operator?
Introduction:
As a quick reminder, the Gel'fand Yaglom theorem uses the generalized zeta-function approach to compute functional determinants of differential operators. Given a differential operator ...
1
vote
0
answers
191
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Polynomials satisfying a three-term recurrence
Let ${p_n}(x) = x{p_{n - 1}}(x) - {a_{n - 2}}{p_{n - 2}}(x)$ for some numbers ${a_n}$ with initial values ${p_{ - 1}}(x) = 0$ and ${p_0}(x) = 1.$
By Favard’s theorem about orthogonal polynomials ...
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4
votes
0
answers
654
views
determinant of fibonacci-sum graphs
We have a simple graph with vertices $\{v_1, v_2, ... v_n\}$.
The adjacency matrix of this graph is $A= (a_{ij})$ so that
$a_{ij}=1$ if $i+j$ belongs to the Fibonacci sequence;
$a_{ij}=0$ ...
- 41
9
votes
2
answers
1k
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When is the determinant a Morse function?
This might be ridiculously obvious, but...
For each $n \in \mathbb{N}$, let $M_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dimensional determinant ...
- 15.5k
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votes
4
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Proving a determinant = 0
The two most elementary ways to prove an N x N matrix's determinant = 0 are:
A) Find a row or column that equals the 0 vector.
B) Find a linear combination of rows or columns that equals the 0 ...
- 51
15
votes
3
answers
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How to show a certain determinant is non-zero
For any $n$ distinct points $x_1,x_2 , \ldots , x_n$ on the real line show that
the matrix $M$ where $M(i,j) = e^{\lambda_j x_i} $ has non-zero determinant
where $\lambda_1 \lt \lambda_2 \lt \ldots \...
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2
votes
1
answer
376
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Endomomorphisms of Chain Complexes of vector spaces and determinants
Let $C_{\ast} : \cdots \to A_{2} \to A_{1} \to A_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$.
And let $f_{\ast} : C_{\ast} \to C_{\ast}$ and $g_{\ast} : C_{\ast} ...
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Morphisms of Spectral Sequences and alternating products
Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences.
Assume that morphisms $...
0
votes
1
answer
842
views
Bounding a determinant ratio
Let $A=[A_{0}\ E;E^{T} \ B]$ be a real positive definite matrix and let $B$ be a principal submatrix. I am interested in tightly bounding $\frac{|B|}{|A|}$ from below in some "explicit" way that will ...
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0
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det(A)det(B) = det(AB+correction), Capelli identities, "factorized" representation of $\mathfrak {gl}_n$
Context: Some probably know that there are Capelli identities which state
$$det(A)det(B) = det(AB+correction)$$ for some matrices with non-commuting elements, they go back to the 19-th century, but ...
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1
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Determinantal formula for the nullspace of a singular matrix
In June 2012, Bill Press and Freeman Dyson published a remarkable paper on the iterated prisoner's dilemma. A key step in their derivation is a simple fact from linear algebra that I feel I should ...
- 78.6k
3
votes
0
answers
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which deformation of a matrix lead to flat deformations of determinantal varieties (fitting ideals)?
Let $(R,m)$ be a complete local ring over a field (of char=0). Consider a (not necessarily square) matrix $A$ over $R$. Consider its fitting ideal, $I_j(A)$. In general, a deformation of the matrix, $...
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2
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a determinantal identity
Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity
$$
\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
...
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votes
1
answer
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Determine if a matrix is unimodular
Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?
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votes
2
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Generalizations of Oppenheim's inequality
The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$.
There has been a lot of beautiful work done ...
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20
votes
1
answer
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When does the $4\times 4$ 'false Sarrus rule' compute the determinant correctly?
This question is most probably not research level, but I thought that the MO folks might like it... Feel free to close.
Here is the motivation: If you have ever taught a maths course for engineers ...
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8
votes
3
answers
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Computing determinants of matrices of linear forms
Suppose we have three $n \times n$ matrices $A$, $B$, $C$ with floating point entries. We would like to compute the polynomial $\det (xA+yB+zC)$. At least in Mathematica, and I think in all computer ...
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Is there a simple relation between the entropy of a matrix and its characteristic polynomial?
Assume $M$ is an invertible positive matrix of rank $N$. The Von Neumann entropy $H$ of a matrix $M$ with eigenvalues $\{ \lambda_n \}$ is
$H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$.
In ...
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0
answers
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Generalized Schur polynomial from block Toeplitz matrices
By using the Jacobi-Trudi identity, one may interpret banded Toeplitz matrices, and minors of such matrices in terms of Schur polynomials, see for example
http://www-stat.stanford.edu/~cgates/PERSI/...
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2
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determinantal identity sought
Suppose $A$ is a $n \times m$ matrix and $B$ is a $m \times n$ matrix. Then it is known that $det(I_{n}+AB)=det(I_{m}+BA)$.
Is there an analogous identity of the form $det(P_{1}+AB)=det(P_{2}+BA)$, ...
- 6,882
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votes
2
answers
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Integral representation of a determinant
In a paper by Mathai, he uses the following integral representation of a determinant,
(or, really, what I give is a simple special case of what he gives), without any explication.
All matrices are ...
- 2,542
7
votes
1
answer
565
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To compute minors of Jacobian of symmetric polynomials
For any $n$ tuple $f_1,f_2,\dots,f_n$ in the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$
one has Jacobian, expressed by the $(n \times n)$-determinants:
$$
J(f_1,\dots,f_n):=|\frac{\partial}{\...
- 446
2
votes
2
answers
239
views
Integer square $2 \times 2$ block matrix inverse
Let $\mathbf{M}$ be an integer square $2 \times 2$ block matrix
$$
\mathbf{M} =
\left(
\begin{array}{cc}
\mathbf{A} & \mathbf{B} \\
\mathbf{C} & \mathbf{D}
\end{array}
\right) ,
$$
where $\...
- 61
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votes
1
answer
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do you know this determinant (basic commutative algebra)?
Let $\ell_1,\dots,\ell_n$ be $d+1$-variate linear forms over complex numbers in variables $X=(X_0,\dots,X_d)$. Consider the $(n-d)$-fold products
$$\ell_{i_1}(X)\ell_{i_2}(X)\dots\ell_{i_{n-d}}(X)=\...
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1
vote
2
answers
203
views
Does integral of A^T(t)*A(t) converge if det(A(t)) does not converge to 0?
Suppose you have an nxn parametric square matrix A(t). I am wondering if I can prove this:
$(lim\_{t\rightarrow\infty}det(A(t)) \ne 0) \Rightarrow (\int^{\infty}A^T(t)A(t)dt$ Does not Converge $)$
- 11
0
votes
0
answers
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Inertia/Gravity in Distance Geometry
The Cayley-Menger Determinant, D(N), slickly calculates the N-dimensional simplex
volume of any N+1 points. One constraint in our 3D world is that D(4)=0.
Give each point a mass (Mi) and dynamic ...
- 1,537
28
votes
4
answers
4k
views
Jacobi's equality between complementary minors of inverse matrices
What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the ...
- 4,354
1
vote
1
answer
328
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 ...
- 11
14
votes
1
answer
1k
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slick-proof-of-trick-for-counting-domino-tilings
The trick for rewriting the number of domino tilings of a simply-connected finite lattice region as the absolute value of the determinant of a matrix (due I believe to Kasteleyn and Percus, but if ...
- 19.4k
5
votes
1
answer
687
views
Characteristic polynomials of certain random symmetric matrices and the complexity of random Morse functions
Investigations concerning random Morse functions led me to the following problem. Consider the classical GOE of $m\times m$ real symmetric matrices $A$ with independent Gaussian entries with ...