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.
2
votes
2answers
81 views
Bijection from $S_{n-1}$ to $\{\sigma \in S_{n} : \sigma(k) = j \}$
Let $n$ be a natural number. Let $k$ be an element of $\{1, \ldots , n\}$. For each j in $\{1, \ldots , n\}$, I want to find a bijection $f_j$ from $S_{n-1}$ to $\{\sigma \in S_n : \sigma(k) = j ...
2
votes
1answer
113 views
complexity cost for which one is greater : determinant or eigen values?
what is complexity cost for determining all of eigen values?
what is complexity cost for calculating determinant ?
2
votes
1answer
44 views
Solving linear equations with Vandermonde
Given this:
$$\begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \vdots & ...
2
votes
1answer
84 views
Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory
In order to proof the following identity:
$$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$
Instead of checking this by brute force, Landau writes de product of ...
2
votes
1answer
73 views
Question on determinants of matrices changing between integer matrices
The following problem came up from a though I had while reading:
Let's say we have $M=\mathbb{Z}^n$ and we have another free $\mathbb{Z}$-module, $N$, inside of $M$ also with rank $n$.
We know we ...
2
votes
2answers
116 views
maximize log determinant subject to a linear constraint
Does anyone know any efficient method to solve the following problem?
$ (\alpha,\beta) = \text{argmax} \log \det (\alpha K_1 + \beta K_2)$
s.t. $c_1 \alpha + c_2 \beta = c_3, \alpha\geq0, \beta\geq ...
2
votes
1answer
40 views
Line integral using variable change
The variable change theorem is the following:
$$\int_B f = \int_A f \circ g \cdot |det\mathcal Jg|$$
So to calculate the following line integral:
$$\int_C(xy)ds$$ where $C = g(t) = (cost, ...
2
votes
2answers
39 views
determinant $s=n+1$
Need help
$A=\begin{vmatrix}
s&s&s &\cdots & s&s\\
s&1&s &\cdots & s&s\\
s&s&2 &\cdots & s&s\\\vdots & ...
2
votes
2answers
108 views
Characteristic polynomial - using rank?
Q:
Let $A$ be an $n\times n$ matrix defined by $A_{ij}=1$ for all $i,j$.
Find the characteristic polynomial of $A$.
There is probably a way to calculate the characteristic polynomial $(\det(A-tI))$ ...
2
votes
1answer
110 views
Number of zero entries in symmetric (0-1)-matrix with full diagonal
Let $S$ be an $n\times n$ symmetric matrix whose diagonal consists only of $1$s and whose other entries are either $0$ or $1$ .
If the determinant and rank of $S$ are known, what can be said about ...
2
votes
1answer
287 views
finding values for determinant to equal 0
I needed to find for which values of $\lambda$ the matrix is singular.
$$
\begin{bmatrix}
1-\lambda & 0 & 3 \\
1 & 1-\lambda & 0 \\
0 & 2 & ...
2
votes
1answer
173 views
Proving determinant product rule combinatorially
One of definitions of the determinant is:
$\det ({\mathbf C})
=\sum_{\lambda \in S_n} ({\operatorname {sgn} ({\lambda}) \prod_{k=1}^n C_{k \lambda ({k})}})$
I want to prove from this that
...
2
votes
1answer
2k views
Determinant of the sum of matrices
Let D be a diagonal matrix and A a Hermitian one. Is there a nontrivial way to calculate the determinant of A from the determinant of A+D and the entries of D?
It can be assumed that the diagonal ...
2
votes
1answer
117 views
How do you show that $\det(A)$ is the product of a linear form in $x_1, x_2, \ldots, x_n$ and a linear form in $y_1, y_2, \ldots, y_n$?
Someone pointed out another solution on how to prove Showing the determinant for a specific type of matrix is $\det(A) = (-1)^n 2^{n-1} \sum_{i=1}^{n} a_1 a_2 \ldots a_{i-1} a_{i+1} \ldots a_n$ and it ...
2
votes
1answer
47 views
Relationship between $|a_{i,j}|$ and $|\alpha^{|i-j|} a_{i,j}|$
What is the relationship between the determinants of the square matrices of equal dimensions $\mathbf{A}$ and $\mathbf{B}$ where each element of $\mathbf{B}$ is equal to the corresponding element of ...
2
votes
1answer
558 views
Generalized variance
Generalized variance is the determinant of correlation matrix. Does increasing the off-diagonal entries (correlation coefficients) decreases the determinant? Is a proof available?
All elements are ...
2
votes
1answer
51 views
Simple/Concise proof of Muir's Identity
I am not a Math student and I am having trouble finding some small proof for the Muir's identity.
Even a slightly lengthy but easy to understand proof would be helpful.
Muir's Identity
$$\det(A)= ...
2
votes
1answer
30 views
Finding Determinants Recursively
From the MIT OCW Linear Algebra (18.06) final exam, question 9:
For square matrices with 3's on the diagonal, 2s on the diagonal
above, and 1s on the diagonal below:
$$A_1=\begin{pmatrix} 3 ...
2
votes
1answer
44 views
Without choosing bases, how to show that the determinant is multiplicative in this sense?
I was recently considering this statement:
Let $V$ be a finite-dimensional $k$-vector space, and let $\phi:V\to V$ be an endomorphism. Suppose that $W\subseteq V$ is a subspace that is stable ...
2
votes
1answer
40 views
Clarifying Theorem 4.11 of Lang's Algebra textbook.
Can someone more explicitly describe Theorem 4.11 in Algebra?
Let $E$ be a module over a commutative ring $R$, and let $v_1,\dots,v_n$ be elements of $E$. Let $A=(a_{ij})$ be a matrix in $R$, and ...
2
votes
0answers
52 views
Matrix inversion is to determinants as matrix logarithm is to what?
I have not put much effort into this question but I have thought about it for a year or so. Is there such thing as a "logarithmic determinant"? The starting point for this is that the determinant of ...
2
votes
0answers
77 views
Determinant, number of non zero columns
Trying to build a reduction from the maximum coverage problem to my research problem, I'm facing this difficulty :
Let $X$ be a $n \times m$ binary matrix (with $m > n$), can we define a square ...
2
votes
0answers
85 views
Intuition in permutations for Laplace Determinant Expansion
Starting with the Leibniz formula for the determinant, I wish to derive the Laplace (Cofactor) Expansion. At the risk of being overly verbose, please see the proof here. Now I understand the idea of ...
2
votes
0answers
92 views
matrix construction
Given any matrix $A$, can one construct a matrix $B$ such that
$B$ is nonnegative and the spectral radius of $B$ is strictly less than 1
the determinant of $A$ is equal to the first entry of $B^*$ ...
2
votes
0answers
100 views
Determinant of the Laplacian of a surface is this correct?
given a surface with metric $ g_{ab} $ i would like to evaluate the functional determinant of the Laplacian in the form
$ - \partial _{s} \zeta (0,E^{2})=\log\det( \Delta + E^{2}) $
then i need to ...
2
votes
1answer
132 views
question in linear algebra, matrices
Given $A$ and $B$, $2\times 2$ matrices, which of the following is necessarily true?
If $A$ and $B$ are both Unitary matrices over $R$ and $\det(A)=\det(B)=1$ then $A$ is similar to $B$.
If $A$ and ...
2
votes
0answers
132 views
Determinants and homomorphisms of general linear groups
Consider the functions $\rho_1:M_1(\mathbb C)\to M_2(\mathbb R)$ where
$$\rho_1(a+bi)=\begin{pmatrix} a&b\\ -b&a \end{pmatrix}$$ and $\rho_2:M_2(\mathbb C)\to M_4(\mathbb R)$ where
...
2
votes
0answers
195 views
Fastest integer matrix determinant software
I need to calculate vast numbers of determinants of integer matrices (size around 30x30 to 50x50) and would like to know the fastest software for this.
It must use exact integer arithmetic as the ...
2
votes
1answer
147 views
Bounding determinants of the following form from above
To bound a determinant of a matrix from above it's quite common to apply Hadamard's inequality. Unfortunately, in the following problem Hadamard's inequality isn't good enough. Are there other methods ...
1
vote
3answers
47 views
Computing the determinant of $\operatorname{id}+aa^t$
What is an easy way to see that $\det(\operatorname{id}_n+aa^t)=1+|a|^2$ for $a\in \mathbb{R}^n$ ?
1
vote
4answers
170 views
Special determinant formula for a specific matrix
How to show that the determinant of the following $(n\times n)$ matrix
$$
\begin{pmatrix}
5 & 2 & 0 &0&0&\cdots & 0\\
2 & 5 & 2 & ...
1
vote
3answers
71 views
Determinant of a $4\times4$ invertible matrix
Let $A$ be a $4$ by $4$ invertible matrix, such that $\det(3A)=3\det(A^4)$.
Then $\det(A)=3$.
Would somebody please give me some clues on this?
Thanks
1
vote
2answers
103 views
How to prove that det($A^{T}A$) is nonnegative?
Why is the determinant of the product of a matrix and its transpose nonnegative?
1
vote
2answers
82 views
Simplest way to calculate a determinant [duplicate]
The big $1$'s here just mean that the lower and upper triangular entries are all $1$'s. The trace entries are all zero. The matrix is for a general $n\times n$ matrix of this form. I'm trying to ...
1
vote
4answers
59 views
The possible number of zero entries in $n\times n$ matrix that would make the determinant non-zero
While preparing an exam, I found the following question:
What is the largest possible number of zero entries in any $5 \times 5$
matrix with a non-zero determinant?
25
15
16
20
...
1
vote
4answers
116 views
Determinant of a matrix $A$ is zero when its has a zero submatrix of dimentions $p \times q$ and …
Let $A$ be a $n \times n$ matrix and suppose $A$ has a zero submatrix of order $p \times q$ where $p + q \ge n+1$. Then $\det(A) = 0$.
I can see this happening when doing Laplace expansion. I can ...
1
vote
1answer
41 views
Determinant is correct but wrong when I try and check it
I have to work out the determinant of the $(n \times n)$ matrix
$$A = \pmatrix{x & y & 0 & 0 &\cdots & 0 \\ 0 & x & y & 0 &\cdots & 0 \\ 0 & 0 & x ...
1
vote
3answers
73 views
If $J$ is the $n×n$ matrix of all ones, and $A = (l−b)I +bJ$, then $\det(A) = (l − b)^{n−1}(l + (n − 1)b)$
I am stuck on how to prove this by induction.
Let $J$ be the $n×n$ matrix of all ones, and let $A = (l−b)I +bJ$. Show that $$\det(A) = (l − b)^{n−1}(l + (n − 1)b).$$
I have shown that it holds ...
1
vote
2answers
83 views
How to workout the determinant of the matrix $D_n(\alpha, \beta, \gamma)$.
I am going through an example in my lecture notes. This is it:
Let's introduce the matrix $D_n(\alpha, \beta, \gamma)$, which looks like this:
$$\pmatrix{\beta & \gamma & 0 & 0 ...
1
vote
5answers
247 views
Suppose A is an n-by-n matrix with its diagonal entries are n and other entries are one. Find determinant of A.
For $n \geq 2$, find the determinant of
$A_{n}=\begin{bmatrix}
n & 1 & 1 &\ldots &1 \\
1 & n & 1 &\ldots &1 \\
1 & 1 & n &\ldots &1 \\
\vdots & ...
1
vote
2answers
65 views
Trace of the matrix power
Say I have matrix $A = \begin{bmatrix}
a & 0 & -c\\
0 & b & 0\\
-c & 0 & a
\end{bmatrix}$.
What is matrix trace
tr(A^200)
Thanks much!
1
vote
1answer
90 views
Prove $\det(A+I)=1$
Need help with my homework.
$A \in M_{nxn}(\mathbb{R})$ is upper-triangular and $A^{n}=0$
Please hint how to prove, that $\det(A+I)=1$
I dont know how it do, know laplace equation
1
vote
1answer
84 views
Determinant of an $n\times n$ matrix with 5's on the diagonal and 2's on the superdiagonal and subdiagonal [duplicate]
Possible Duplicate:
Special determinant formula for a specific matrix
How to find $\det A_n$ as a function of $n$?
$$A_n=\begin{pmatrix} 5&2 &0& 0 & \ldots & 0\\
...
1
vote
2answers
136 views
Row Reduction with Cofactor Expansion
My calculator says the determinant of $$\begin{pmatrix}3 &0&6&-3\\0&2&3&0\\-4&-7&2&0\\2&0&1&10\end{pmatrix}$$ is $396$. However, the website I got the ...
1
vote
2answers
149 views
Determinant of a big matrix
I have done a program to calculate a determinant of a matrix. My program works, but the problem is that it takes long time to calculate, especially for big matrix. Could you tell me how can a perform ...
1
vote
1answer
117 views
determinant of an $ n\times n$ matrix type [duplicate]
Possible Duplicate:
How to calculate the following determinants
Computing determinant of a specific matrix.
How can one compute the determinant of an $n\times n$ matrix where all the ...
1
vote
2answers
64 views
Matrix Identity Proof
Let $A$ and $C$ be $3 \times 2$ matrices and let $B$ be a $2 \times 2$ matrix such that $AB=C$. Prove that:
$$||A_1 \times A_2 || \cdot |\det B| = ||C_1 \times C_2 ||$$
where $A_i$ and $C_i$ are the ...
1
vote
1answer
34 views
Is it true that in $Mat(n,n) $ the set of singular matrices forms a hyperplane?
Is it true that in $Mat(n,n)$ the set of singular matrices forms a hyperplane, separating the matrices of positive determinant from the matrices of negative determinant?
This is my intuition, but ...
1
vote
1answer
56 views
Calculating determinant of a matrix whose non-zero elements are two sub-matrices on its diagonal
Given a $m \times m$ square matrix $M$:
$$
M = \begin{bmatrix}
A & 0 \\
0 & B
\end{bmatrix}
$$
$A$ is an $a \times a$ and $B$ is a $b \times b$ square matrix; and of course $a+b=m$. All the ...
1
vote
2answers
72 views
Determinant of a block matrix with $\mathrm{Id}$ and $0$ in the diagonal
How to compute the determinant $\det A$ depending on $B$ and $C$, where
$$ A = \left(\begin{matrix}\mathrm{Id} & B \\ C & 0 \end{matrix} \right), $$
a) when $C$ is square,
b) $C$ has more ...
