0
votes
1answer
20 views

How do i prove the Leibniz formula of the determinant over a commutative ring?

Let $R$ be a commutative ring. My definition for the determinant over $M_n(R)$ is defined inductively as $\det_{n+1}(A)=\sum_{j=1}^n (-1)^{1+j}A_{1j} \det_n(\tilde{A_{ij}})$. (Here, $(-1)$ denotes ...
8
votes
2answers
257 views

Determinant of the linear map given by conjugation.

Let $S$ denote the space of skew-symmetric $n\times n$ real matrices, where every element $A\in S$ satisfies $A^T+A = 0$. Let $M$ denote an orthogonal $n\times n$ matrix, and $L_M$ denotes the ...
1
vote
0answers
42 views

Complex matrices: looking for homomorphism

Let $\mathbb{C}$ denote the complex numbers, and let $M_2(\mathbb{R})$ be the ring of $2$ by $2$ matrices with real entries. Define a function $f:\mathbb{C} \to M_2(\mathbb{R})$ by $ f(a+bi) = ...
1
vote
0answers
18 views

Preservation of determinants mod some ideal

Given a matrix with entries drawn from some field or commutative ring, what are the conditions for the determinant to be preserved mod some ideal? For a concrete example, I am thinking of matrices ...
0
votes
1answer
41 views

What is the order of this group? [duplicate]

Let $H$ be the subgroup of the group $G$ of all $2 \times 2$ non-singular matrices whose entries are integers modulo a given prime $p$ consisting of those and only those matrices in $G$ whose ...
4
votes
4answers
221 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
3
votes
2answers
74 views

how to compute the determinant of a linear map

Let $V$ be the vector space of $m\times n$ matrices over a field $F$. Fix an $m\times m$ matrix $A$ and an $n\times n$ matrix $C$, and consider the map $\phi: V\longrightarrow V$ defined by ...
4
votes
1answer
52 views

If $A\in M_n(R)$ and $\det A$ is not a zero divisor, what can we say about its entries?

I am working on this proof and think I have a lemma that will get it for me. However I am not sure if this lemma is true and can not figure out how to prove it, if it is. Here goes Given some $A\in ...
1
vote
1answer
67 views

Equality of discriminants of integral bases (statement in Ireland and Rosen, A Classical Introduction to Modern Number Theory)

I'm doing independent study and need assistance. This is taken from Ireland and Rosen's A Classical Introduction to Modern Number Theory, Chapter 12. Let F/Q be an algebraic number field, D the ring ...
4
votes
0answers
122 views

What does abstract algebra have to say about the determinant?

The determinant is a homomorphism from the multiplicative monoid of matrices to the multiplicative monoid of a field (right?). I find this to be the most intuitive way to interpret some of the ...
6
votes
1answer
141 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
2
votes
4answers
224 views

Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$

Let $V$ be a real inner product space and $S=\{v_1,v_2, \dots, v_n\}\subset V$. How am I to prove that $S$ is linearly independent if and only if the determinant of the matrix $$ ...
1
vote
1answer
197 views

REVISITED$^1$: How is the determinant of a matrix $A\in M_{2\times 2}(\mathbb{R})$ considered a bilinear form?

I'm trying to prove that $B(X,Y)=\det (X+Y) - \det (X) - \det (Y)$ is a blinear form on the vector space $A$ is from, and also trying to determine if it is an inner product space. I think if I know ...
2
votes
1answer
559 views

Column entries of a matrix sum to zero, so what are the properties?

What kind of properties does a matrix whose column entries sum to zero have? $$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & ...
0
votes
3answers
122 views

$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form

Prove that every $n$-linear alternating form on a vector space of dimension less than $n$ is the zero form.
3
votes
1answer
93 views

Find the smallest square matrix in which some objects fit following some rules

I have to put some objects in a matrix. The data of these objects is given in another matrix in which each line contains an object, and the first column represents its width, and the second its ...
14
votes
2answers
173 views

Is a linear combination of minors irreducible?

Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
2
votes
1answer
98 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
57 views

Is $N_{k\subset K}$ the only *norm* on the field extension $k\subset K$?

In several examples of field extensions the norm function is very useful. For instance, in $\mathbb{Q}\subset\mathbb{Q}(\sqrt{2})$, the norm is $N(x+y\sqrt{2})=x^2-2y^2$. In ...
6
votes
1answer
239 views

Do multiplicative maps of matrices factor through determinants?

Given a map $f:M_n(k)\to k$ (with $k$ some field) such that $f(AB)=f(A)f(B)$ for all matrices $A$ and $B$, is it necessarily the case that $f$ factors through the determinant, i.e. does there exist a ...
8
votes
2answers
460 views

Elementary proof that if $A$ is a matrix map from $\mathbb{Z}^m$ to $\mathbb Z^n$, then the map is surjective iff the gcd of maximal minors is $1$

I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by an $m \times n$ matrix $A$, then the map is surjective iff the gcd ...
4
votes
1answer
754 views

Is there a formula for the determinant of the wedge product of two matrices?

I was going over the Wikipedia page for exterior products of vector spaces and we can define the determinant as the coefficient of the exterior product of vectors with respect to the standard basis ...