# Questions tagged [linear-algebra]

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

3,659 questions
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### A random walk matrix has eigenvalue 1 with multiplicty 1 - why?

A random walk matrix has largest eigenvalue 1 with multiplicty 1 - why? Let $G$ be a non-directed, regular connected graph with degree $d$. Let $A$ be its random walk matrix, i.e. it's adjacency ...
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### Definition of inner product for vector spaces over arbitrary fields

Is there a canonical definition of the concept of inner products for vector spaces over arbitrary fields, i.e. other fields than $\mathbb R$ or $\mathbb C$?
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### factorization of the product of a matrix element and its cofactor

Hi, this is kind of continuation of this thread to concentrate on a specific problem from linear algebra and analysis that, I think, is rather interesting for itself. Here we go: 1) Main problem: ...
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### How to compute the rank of a matrix?

Okay, that's a misleading title. This is a somewhat subtler problem than undergraduate linear algebra, although I suspect there's still an easy answer. But I couldn't resist :D. Here's the actual ...
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### Is there a name for the matrix equation A X B + B X A + C X C = D?

I happen to be working on a problem that reduces to solving the following equation: $$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$ where A through D are known matrices ( A, B, D ...
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### How can I learn about doing linear algebra with trace diagrams?

There is a wikipedia article. There is a paper by Elisha Peterson. I tried reading these but they don't seem to click for me. Are there books or other resources for learning how to do linear ...
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### Ways to Synthesize Topics in Linear Algebra

Hello, I am currently studying linear algebra right now. In general, the material is pretty straight-forward but it doesn't seem particularly interesting. I suppose that the main thing that I am ...
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### Notions of Matrix Differentiation

There are a few standard notions of matrix derivatives, e.g. If f is a function defined on the entries of a matrix A, then one can talk about the matrix of partial derivatives of f. If the entries of ...
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### Is there a specific name for matrices with nonsingular principal submatrices?

Is there a specific name for matrices with nonsingular principal submatrices?
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### How do you construct a symplectic basis on a lattice?

Is this possible to do constructively? The only sources that I have for the possibility of this construction is an exercise in Lang's Algebra (on p. 598, I believe) which states that one can be ...
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### What is the weakest condition on the matrices A_k that guarantees v_k->0 => A_kv_k->0 ? [closed]

What is the weakest condition on the sequence of real matrices A_k that guarantees that whenever a sequence of real vectores v_k converges to zero, the product A_kv_k also converges to zero? Edit: ...
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### Graphs with incidence matrices whose pseudoinverses are proportional to their transposes

When I was working on my PhD dissertation, I came across a physical situation involving nodes and flows between them. It turned out that I was working with a complete oriented graph $K_n$ (all nodes ...
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### In an n-dimensional linear 2nd-order ODE, why is the transpose-inverse to a system of solutions also a solution?

I'm at a sticky spot in my research. Namely, I have a particular fact, and it ought to have a short proof, but the only way I know how to show it is long and drawn out, and I don't like it and worry ...
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### Hermitian matrices with prescribed number of positive and negative eigenvalues

Let $H$ be a linear subspace of the space of Hermitian $n\times n$ matrices. Is there a good characterization of those $H$ such that every $A\in H$ has at least $k$ positive and $k$ negative ...
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### When to pick a basis?

Picking a specific basis is often looked upon with disdain when making statements that are about basis independent quantities. For example, one might define the trace of a matrix to be the sum of the ...
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### Is it that only with normal matrices, the transition matrix to its [del: inherent] [ins: own] basis is unitary?

Does this even make sense what I translated into english? PS. I am probably gonna delete this question eventually
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### Eigenvalues of matrix sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite? I am ...
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### What are the components of a transpose operator from $\mathbb R^{n\times n}$ to $\mathbb R^{n\times n}$?

Say I'm working in the space of linear transformations from $\mathbb R^n$ to $\mathbb R^n$ and I've picked a basis so I can identify with any operator a component matrix in $\mathbb R^{n\times n}$. ...
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### Shear transformations

Where can I learn more about shear matrices? The Wikipedia article is not enough, and sadly it does not have any references. I understand they are linear transformations. Do they form a group? How ...
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### Bivectors in 3 and 4 dimensions

The big questions behind are: Is a bivector a two-form? Why a bivector is simply a vector in 3 dimensions? How to distinguish between vectors and bivectors in 3D? Why all bivectors are not vectors ...
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### When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?

Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...
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### Quantifying Aggregate Vector Strength/Vector Arithmatic

Say I have 5 vectors and I measure the similarity of each one to a fixed reference vector using cosine similarity. But now what I want to do is understand the aggregate or collective strength of these ...
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### Generators for congruence subgroups of SL_2

For positive integers $n$ and $L$, denote by $SL_n(Z,L)$ the level $L$ congruence subgroup of $SL_n(Z)$, i.e. the kernel of the homomorphism $SL_n(Z)\rightarrow SL_n(Z/LZ)$. For $n$ at least $3$, it ...
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### Matrices into path algebras

I was thinking about quivers recently, and the following idea came to me. Let ei,j denote the matrix unit in Mn for 1 ≤ i,j ≤ n. Let Γ denote the complete quiver on vertices {1, …,...
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### Abelianization of Lie groups

If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...
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### elementary Ext^1 intuition

I am wondering what sort of basic basic intuitive meaning Ext1(M,N) has. As a base case: if <...
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### Which computer algebra system should I be using to solve large systems of sparse linear equations over a number field?

This is related to Noah's recent question about solving quadratics in a number field, but about an even earlier and easier step. Suppose I have a huge system of linear equations, say ~10^6 equations ...
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### Question about orthogonal matching pursuit

Let y be a n-vector, X a n-by-p matrix of full rank (p < n) and b a p-vector, so that y = Xb + e, for some noise vector e. I am not sure how to show reduction of error in orthogonal matching ...
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### Variation on a matrix game

The original problem appeared on last year's Putnam exam: "Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008×2008 array. Alan plays ﬁrst. At each turn, ...
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### Infinite matrices and the concept of “determinant”

Suppose we have an infinite matrix A = (aij) (i, j positive integers). What is the "right" definition of determinant of such a matrix? (Or does such a notion even exist?) Of course, I don't ...
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### Orbits of real groups, canonical forms of matrices

There are a lot of results in textbooks concerned with canonical forms of matrices under certain complex groups of transformations, e.g. GL(n|C), O(n|C),... Could anybody give me references where the ...
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### Lifting bases for (Z/pZ)^n to Z^n

The following question came up in my research. I suspect that it has a slick answer, but I can't seem to find it. Fix an integer n>=2 and a prime p. Define X(n) to be the set of primitive vectors ...
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### How do you rotate a matrix to maximum sparsity?

Given a matrix M, I want to find an orthogonal matrix U that maximizes the number of entries that are zero in the product MU. How do I go about doing this?
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### Is the center of a free (as a module) algebra free?

A submodule of a free module need not be free (for instance, in the free Z[X]-module Z[X] the submodule generated by 2 and X is not free). But over a principal ideal domain, submodules of free modules ...
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### Change of basis with Multilinear fucntion [closed]

Take a multi-linear function(or functional) M that takes m arguments V1…Vm, each with a dimension n. Consider only the case where m=n. Let there be a change of basis performed on the arguments(V1...Vm)...
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### What is the size of the category of finite dimensional F_q vector spaces?

The size of a finite skeletal category C in the sense of Leinster is defined as follows: Label the objects of C by integers 1,2,...,n and let aij be the number of morphisms from i to j (for i and j ...
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### Friedberg, Insel, and Spence Linear Algebra example

In the chapter 6.4 on normal and self-adjoint operators, there is an example of an infinite dimensional inner product space H that has a normal operator but that has no eigenvectors. The space is the ...
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### Conjugation in SU(2)

For any two matrices $P,Q \in SU(2)$, with $tr(P)=tr(Q)=0$, does there always exist some $G\in SU(2)$ such that $G P G^{-1} = -P$, and $G Q G^{-1} = -Q\ ?$
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### An “existence contra partition of unity” statement for integer matrices?

While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind. Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
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### Linear Algebra Over $F_{2}$

Suppose we call a subset S of $F^{n}$ ($F$ is the field with two elements) good if for any $x$ and $y$ (possibly $x=y$) we have $[x,y]=1$ where $[ , ]$ denotes the obvious bilinear form on F. What's ...
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### Non-conjugate words with the same trace

Let n>=2, p a large prime, G = SL_n(Z/pZ). If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...
It seems "common knowledge" that the following holds: Let $T$ be a linear transformation on nxn matrices with complex coefficients that preserves the determinant. Then there exists matrices U and V ...