3
votes
2answers
42 views

“Sandwich theorem” for eigenvalues of symmetric matrices

I am looking for a reference for the following result for symmetric matrices Let $A\in\mathbb R^{n\times n}$ be symmetric with eigenvalues $\lambda_n \leq\ldots\leq\lambda_1,\, M\subset \lbrace ...
0
votes
1answer
38 views

On the decomposition of Stochastic matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
2
votes
1answer
13 views

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$, how to call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$ How do people call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$? Sum of positive components of $X$? The positive semi definite part of $X$? ...
1
vote
1answer
36 views

Approximate Equivalent To Michael Spivak's text, “Calculus” but for Linear Algebra?

Does anyone know of an approximate equivalent To Michael Spivak's text, "Calculus" but for Linear Algebra? I love the way this book is written! It is simultaneously rigorous and thorough without ...
0
votes
0answers
8 views

Reference request: result concerning Leray trace

Let $V$ be a vector space (possibly of infinite dimension). For a linear homomorphism $f\colon V\to V$ define $$N(f)=\bigcup_{n\in\mathbb{N}} \operatorname{ker}(\underbrace{f\circ\ldots\circ ...
0
votes
0answers
21 views

Extending a trace on algebra to a trace of systems of algebras

Suppose, we have a trace $\tau$ on some algebra $\mathcal{A}$, i.e. $$\tau(aA+bB)=a\tau(A)+b\tau(B)\ \forall A,B\in\mathcal{A}, \forall a,b\in\mathbb{C}$$ The question rises, what are then the ...
1
vote
4answers
134 views

Suggestion for a book on Linear Algebra [duplicate]

Please suggest a Linear Algebra book with an introduction and rigorous theory (description) on Eigenvectors , eigen-values , Cayley-Hamilton theorem , Diagonalisation of matrices ; Quadratic forms ( ...
0
votes
0answers
29 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
1
vote
1answer
36 views

References for coordinate-free linear algebra books

I'm looking for some (or one) good book(s) that teach linear algebra either purely coordinate-free or ones that present the standard bag-of-tools alongside coordinate-free alternatives or discussions. ...
0
votes
1answer
22 views

Pseudo-Theorem on parallel lines in a quadrilateral - Proof & Reference Request

Yesterday I asked a question concerning an argumentation, and keep on thinking about the problem I realized that what I was missing is probably a basic result in linear algebra. Actually I am not sure ...
5
votes
0answers
84 views

Reference Request: Prereqs for Lecture Notes on “Abstract Linear Algebra”

I just found this set of lecture notes on linear algebra which seems to go over several things I've been wondering about as I study linear algebra. Unfortunately there are very few exercises in the ...
1
vote
2answers
82 views

Linear Algebra book supplement to Axler

I have been self-studying Linear Algebra from Linear Algebra done Right by Axler for the past one month. So far I haven't encountered Matrices/solving linear equations and the book doesn't seem to ...
0
votes
0answers
42 views

Book Recommendation for Understanding Linear Algebra [duplicate]

I'm a Computer Science student and I took an elementary level Linear Algebra course. I realised that I can't really understand many topics, especially the linear transformations deep enough. I'm ...
1
vote
1answer
35 views

Matrix calculus - derivative of scalar function of matrix function of scalar

Cross posted from stats.stackexchange as it wasn't getting any love (given it a few days)! In the context of REML estimation there is the result (ignoring some constants) that (my interest is in the ...
1
vote
1answer
51 views

Trace of the exterior power as a determinant

Let $A$ be a matrix. According to Wikipedia, $$tr(\wedge^k A) = \frac{1}{k!} \det \begin{pmatrix} tr (A) & k-1 & 0 & \cdots \\ tr (A^2) & tr (A) & k-2 & \cdots \\ \cdots & ...
1
vote
0answers
21 views

Reference request for topic “Change of coordinate matrix”

I recently started with vector spaces. I understood every theorem from the course book uptil isomorphism. Then, i saw this particular topic and was totally blank. I don't know whether it was the ...
0
votes
1answer
15 views

How to test if $m$ vectors are linearly dependent when they are $n$ dimensional and $m < n$

I'll be shocked if this isn't a duplicate, but I haven't had a lot of luck finding an answer to this so far. How do you test if a set of vectors $v_1, \ldots v_m \in \mathbb{R}^n$ are linearly ...
0
votes
2answers
70 views

How to understsand eigenvalues and eigenvectors.

I know basic linear algebra (what is a matrix, what is a determinant, what is a square matrix, what is an inverse of a matrix, how to add/sub/multiple matrices etc.) But I am finding the concept of ...
1
vote
3answers
62 views

Inner Product Spaces, suggestion for book.

Can you suggest me name of some books which would help me visualize IPS better? Like, books having diagrams and stuff?
0
votes
0answers
20 views

Largest average principal submatrix of a symmetric matrix.

I am wondering if there exists literature on the following problem: Let $X$ be an $n \times n$ symmetric matrix. How do you identify the $k \times k$ principal submatrix of $X$ with the largest ...
13
votes
8answers
506 views

Very good linear algebra book.

I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), ...
1
vote
1answer
43 views

Existence and construction of asymmetric codes

By a $[n,k]_2$-code I mean a $k$-dimensional $\mathbb{F}_2$-subspace of $\mathbb{F}_2^n$. Such a code $C$ admits a symmetry $\sigma \in S_n$ if for any word $w \in C$ we also have $w^\sigma \in C$, ...
1
vote
1answer
23 views

Linear Optimization Study Material

I've recently enrolled in a linear optimization course, and it's been a while since I've taken linear algebra. I do not yet have access to the book for the course or I would skim it to see what I need ...
4
votes
2answers
34 views

Regular matrices references

Can someone suggest me a book or a lecture note which covers regular matrices with all theories related to it? Any assistance will be much appreciated. (By regular I mean some power of the matrix is ...
1
vote
3answers
124 views

Choosing good textbooks in linear algebra, analysis and graph theory

I need some advices to choose good undergraduate textbooks in LINEAR ALGEBRA, ANALYSIS and GRAPH THEORY. I found: Gilbert Strang // Introduction to Linear Algebra - Welleslay Cambridge Press (2009) ...
0
votes
0answers
13 views

Reference request: Tensor Product of $m$ $SU(N)$ algebras

I'm working on quantum mechanics of linear $SU(N)$ chain of sites. Specifically, I would like to study a Tensor product of $m$ Lie algebras $\mathfrak{s}\mathfrak{u}(N)$ $$ V\in SU(N)\\ W = ...
2
votes
1answer
43 views

Is every element of a complex semisimple Lie algebra a commutator?

Let $L$ be a (finite-dimensional) complex semisimple Lie algebra. Then we know that $L = [L,L]$. Is it true that every element of $L$ must be a commutator? Since a complex semisimple Lie algebra is ...
3
votes
1answer
61 views

categorification and linear algebra

Vect is the category with objects as vectors and arrows as linear transformations between them. Then these arrows have quite a bit of structure. We can take the transpose, trace, determinant, ...
0
votes
1answer
27 views

Sylvester domains

I'm an undergrad mathematics student and I'd like to request some books about Sylvester domains. Specifically I'd like to understand the fact that not all modules are Sylvester domains. I just proved ...
2
votes
5answers
366 views

Book recommendation for Linear algebra.

I am looking for suggestions, it has to be a self study book and should be able to relate to applications to real world problems. If it is more computer science oriented , that would be great.
0
votes
1answer
56 views

Matrix representation of complex numbers in exponential form

Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ...
3
votes
2answers
47 views

Arbitrary (i.e. not necessarily finite-dimensional) vector spaces; reference request.

Its virtually impossible to complete an undergraduate degree these days without studying finite-dimensional vector spaces in quite some detail. So like most of us, I've done all that; however, just ...
0
votes
0answers
16 views

Books in spectral theory for finite dimensional spaces

I'm looking for beginner books of spectral theory for finite dimensional spaces. I've already heard about this subject, but I don't know where I can find it. What's the domain of this subject? (Linear ...
1
vote
0answers
33 views

Matrices of the form $A^p=(a_{ij}^p)$

I am wondering if there is a name for these kind of matrices and if they are interesting or not? Do they even exist? Let $A$ be a $n\times n$ matrix with elements $a_{ij}$. $A= (a_{ij})_{i,j\in\{1, ...
1
vote
5answers
61 views

Why if $aX+c=bX+d$ then $a=b$ and $c=d$?

There is theorem in linear algebra. I forgot it!! But I remember something from it. Can you please give me a reference? It is related to something like this. If I have two polynomials ...
1
vote
1answer
45 views

Hermitian matrix the only diagonizable

During the last lecture one of my professors claimed that the hermitian matrix is the ONLY complex matrix which was diagonizable. This seems strange to mee (not to say a very very strong claim to ...
5
votes
1answer
132 views

iterated dual vector spaces

Let $K$ be a field and $\mathcal U$ a universe such that $K\in\mathcal U$. (Here, "universe" means "uncountable Grothendieck universe".) Let $\mathcal C$ be the category of $K$-vector spaces belonging ...
3
votes
2answers
135 views

Smooth spectral decomposition of a matrix

Let $A : x \mapsto A(x)$ be a $C^\infty$ map from the half-plane $\left\{ (x_1,x_2,\cdots,x_n) \in \mathbb{R}^n,\ x_n>0\right\}$ to the space of symmetric matrices with real coefficients. Suppose ...
1
vote
1answer
25 views

Some terminology and reference questions on singular values

Let $T: V \rightarrow W$ be an operator between to inner product spaces. Then singular values $s_1 \leq s_2 .... \leq s_n$ of $T$ are square roots of eigenvalues of $T^*T$ where $T^*$ is the conjugate ...
0
votes
0answers
14 views

Reference for linear algebra in a more generelized way

I'm studying the chapter VII in the classical Hungerford's algebra book and he treat matrices in a very generalized way (with entries in a ring with unit) and their relationship with ...
1
vote
1answer
57 views

Where can I find good examples about Algebra (but not only): Usual counter-examples, but also limit cases, rare ones, etc [duplicate]

I recently discovered the importance of examples and couter-examples in mathematics. Where could I find good examples books or anything related to it ? I am particularly looking for rare limit-cases, ...
2
votes
2answers
148 views

Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
3
votes
1answer
151 views

Hard problems book in linear algebra

Could you suggest me a book where I can find hard problems in Linear Algebra for an undergraduate student? Thanks in advance.
0
votes
0answers
24 views

Reference Request: Matrix of a composition(sum) of two operators is the Kronecker product (sum) of the matrices of each operator

Here is link to an example: http://en.wikipedia.org/wiki/Kronecker_sum_of_discrete_Laplacians, but it provides no references. Hoffman and Kunze(Linear Algebra) develop linear algebra with explicit ...
6
votes
0answers
155 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
1
vote
2answers
231 views

What are the best texts on undergraduate linear algebra?

I have recently finished a course in 'elementary linear algebra,' which entails basic systems of linear equations, in-depth study on matrices, the basics of vector space, inner product spaces, linear ...
1
vote
0answers
54 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
0
votes
0answers
44 views

Quadric surfaces classification

I have an exam soon on quadrics. However, I don't have enough exercises, especially ones involving finding standard form of quadrics using affine and orthogonal transformations - rotation, translation ...
0
votes
0answers
21 views

Is it possible to group linear maps by similarity or likelihood that they are identical?

Given a set of linear maps $f: x \rightarrow y$, is there a way to (statistically or otherwise) determine how likely two linear maps are identical even if I do not have enough data to determine $f$? ...
0
votes
1answer
141 views

Most “beautiful” presentations of the basic proofs for vector spaces?

I am familiar with the standard proofs presented in textbooks for stuff like linear independence/dependence, the dimensions of common vector spaces, any basis for a vector space V must be linearly ...