0
votes
1answer
28 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
41 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 ...
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
57 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
39 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 ...
4
votes
1answer
100 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
123 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
0answers
17 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
49 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
77 views

Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
2
votes
1answer
88 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
15 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
121 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
102 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
40 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
33 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
107 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 ...
3
votes
2answers
44 views

Proof that diagonally dominant matrices are regular - Reference request

I know that it is easy to proof that diagonally dominant matrices are regular (non-singular) by the gershgorin circle theorem. But the theorem that diagonally dominant matrices are regular was ...
3
votes
1answer
68 views

The best place to excercise Linear Algebra or Calculus?

Could anyone tell me some good website for excercising Linear Algebra or Calculus? Thank you very much for every advice :)
0
votes
1answer
23 views

Fractional and irrational matrix powers

What is a good reference to learn the basics about raising a matrix to a rational power or an irrational power? So I am interested in the existence and computation of things like $A^{\frac{1}{3}}$ or ...
1
vote
0answers
37 views

Reference for Codimension in Infinite Dimensional Normed vector spaces

There are a couple identities I would like to use related to the codimension and its relationship to the annihilator; some of these seem to be true for all normed vector spaces, and others seem only ...
0
votes
0answers
42 views

MIT Lectures of `Applied mathematics` - a reference request.

I'm translating this information directly from the official course syllabus, So I hope I'll do it right. I'm a computer science student and currently I'm learning course called ...
0
votes
0answers
50 views

When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
1
vote
2answers
43 views

Expressions for Permanent of a Matrix

Given that the permanent of a matrix can be written in a similar form as the determinant, as a sum of permutations of the elements of the matrix, is there also a relationship between the permanent and ...
1
vote
2answers
145 views

Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
1
vote
0answers
44 views

Linear algebra, affine space, and floor function

My question is mostly: is there a name for this kind of things. I am mostly interested by finding book or articles about what follows, but without even a word or a name, it is quite hard to search for ...
1
vote
3answers
58 views

Reference on constructing preconditions (beginner level)

I took linear algebra course this semester (as you've probably noticed looking at my previously asked questions!). We had a session on preconditioning, what are they good for and how to construct them ...
1
vote
0answers
28 views

book related query

I have been solving a lot of problems in algebra, calculus, probability and statistics. But have never encountered problems that consist of every mathematical field mentioned above (at max two ...
2
votes
0answers
77 views

Higher dimensional Euclidean geometry problem

In my engineering/physics research, I am facing one math problem which I believe should be well established in mathematics... I have a linearly spanned space given by the column vectors of the ...
1
vote
4answers
259 views

Book Recommendations for Linear Algebra Proofs

I'm taking a graduate Linear Algebra course and have limited experience writing proofs (mostly from a discrete math class). Can anyone recommend good books to teach you how to write proofs for linear ...
-1
votes
1answer
131 views

A Book for Linear Algebra [duplicate]

I want to start learning Linear Algebra, I have no background about this subject except high school mathematics that doesn't includes complex number and matrices. I found the following books: ...
0
votes
0answers
15 views

References for incompatible systems and least squares problem

I need some not-so-complicated references on the subject "incompatible systems and least squares problem" to understand the concept better and deeper. My course book is linear algebra and optimization ...
3
votes
4answers
143 views

Elementary linear algebra sources

I took matrix computations course, our course book is Numerical Linear Algebra and Optimization. As a computer science student, sometimes I get the impression that I lack some fundamental background ...
3
votes
2answers
169 views

Reference request: trace integral formula

This question is not about the proof but about a reliable source where one can find the following formula for the normalised trace $\mbox{tr}$ of a complex $(n\times n)$-matrix: $$\mbox{tr}(A) = ...
4
votes
0answers
78 views

Can the “inducing” vector norm be deduced or “recovered” from an induced norm?

Can the "inducing" vector norm be deduced or "recovered" from an induced (operator) norm? This question occurred to me after seeing this question. I'm hoping that perhaps there exists something like ...
-1
votes
4answers
116 views

Linear Algebra and Set Theory book recommendations.

I would like to studying linear algebra and set theory. Does anyone have a a good recommendation of books/resources/etc.?
4
votes
1answer
120 views

Dual and adjoint operator

Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual ...
2
votes
3answers
215 views

Roadway and book recommendations to math study.

I had some calculus, linear algebra and complex analysis courses back in college. But it is not comprehensive. And I felt that my college math was not taught in a logical sequence (maybe because my ...
2
votes
1answer
113 views

How to translate between differential forms and tensor index notation

The books on Manifold theory & geometry that I studied introduce connection and curvature in the language of differential forms. But Physics books on the other hand like (General Relativity by ...
3
votes
4answers
119 views

What's a good reference to study multilinear algebra?

This semester I'm taking a course in linear algebra and now at the end of the course we came to study the tensor product and multilinear algebra in general. I've already studied this theme in the past ...
1
vote
0answers
115 views

Determinant of symmetric block matrix

I have a block matrix of the form: $B = \left[\begin{array}{cccc} A_1 & C & \dots & C\\ \vdots & A_2 & \dots & C\\ C & \vdots & \ddots & \vdots\\ C & C & C ...
1
vote
1answer
96 views

Suggestions for comprehensive maths book library

I've problem that I'm slowly forgetting the math I've learned in early years at university (right now I'm in final year of Mgr. degree as theoretical physicist). I'd like to assemble a finite but ...
0
votes
0answers
96 views

The ith leading principal submatrix obtained by interchanging columns

Let $S \in GL_n(\mathbb{Q})$ be a non-singular symmetric $n\times n$ matrix with LDU decomposition. Let $L^T=U=(f_{ij})$ where $f_{ii}=1$ for all $i = 1, \ldots, n$ and $f_{ij} = 0$ if $ i > j$, ...
2
votes
0answers
42 views

Reference request on pseudo-determinants

I am looking for a reference on pseudo-determinants$^{(1)}$. I am mostly interested on general and/or basic equalities and properties such as those obtained for determinants. Any pointers would be ...
1
vote
1answer
140 views

Intermediate textbook in Linear Algebra

I am looking for a Linear Algebra textbook that for those who just finished elementary Linear Algebra. I just finished Introductory to Linear Algebra by Strang and read to Least Squares and ...
3
votes
1answer
351 views

Is this reading path recommended?

Since doing math requires learning it first, I 've chosen a series of books to understand some ''Higher math''(which I want to read over a period of several years),and would like to see some ...
1
vote
0answers
101 views

Rank-2n tensor algebra eigenvalue equation

Im interested in resources and work done on the eigenvalue equation for rank-2n tensors: $$ M_{ij}A_{j} = \lambda A_{i} \\ $$ $$ M_{ijkl}A_{kl} = \lambda A_{ij} \\ $$ $$ M_{ijklmn}A_{lmn} = \lambda ...
1
vote
1answer
78 views

Study of Matrix Calculus

I need to study matrix calculus such as integration, differentiation, differentiation of functions of determinants and inverse matrices and then also other matrix based calculations such as ...