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.

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 ...
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 ?
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 & ...
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 ...
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 ...
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 ...
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, ...
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 & ...
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 $(\det(A-tI))$ ...
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 ...
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 & ...
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 ...
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 ...
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 ...
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 ...
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 ...
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)= ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $B^*$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $a\in \mathbb{R}^n$ ?
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 & ...
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
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 & ...
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!
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
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\\ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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