Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Fast way to inverse B'CB+D

$\mathbf {A = B'CB}$, where $\mathbf A$ is of dimension $n \times n$, $\mathbf C$ is m by m, positive definite and symmetric, $\mathbf B$ is of dimension $m \times n$, and $n >> m$. Inversion ...
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Variable bounds of under-determined linear system

If I have a non-negative, under-determined linear system $\mathbf{Ax}=\mathbf{b}$ $\mathbf{x}\geq\mathbf{0}$ is there a fast way to compute the upper and lower bounds on values of each element of ...
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27 views

Equality case in the Frobenius rank inequality

In many linear algebra books, the following rank inequalities are found: Frobenius inequality Let $A$, $B$ and $C$ be three matrices such that the product $ABC$ is defined. Then ...
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2answers
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Jordan chain when matrix has only one eigenvalue.

A $12\times 12$ matrix has sole eigenvalue $3$. It is given that the kernels of $A-3I$, $(A-3I)^{2}$, $(A-3I)^{3}$ and $(A-3I)^{4}$ have dimensions $4$, $7$, $9$ and $10$ respectively. What ...
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43 views

Closure and compactness of the set of real eigenvalues ​​of a real matrix.

Let $A$ be a part of $\mathcal{M}_n(\Bbb{R})$ and $B$ the set of real eigenvalues ​​of the matrix $A$. 1) Show that if $A$ is compact then $B$ is compact as well. 2) If $A$ is closed ...
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$\{x: x^T Ax = a\}$ is unbounded for any $a \in \mathbb{R}$, then A is indefinite

Let $A_{3\times 3}$ be symmetric matrix. If set $\{x: x^T Ax = a\}$ is unbounded for any $a \in \mathbb{R}$, then A is indefinite. Let's take $a=1$. We are told that there is at least 1 x ...
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31 views

Equality in the Collatz-Wielandt-formula

Let A be a matrix with positive entries. The perron-frobenius-theorem states that A has a positive dominating simple eigenvalue, called the perron-frobenius-eigenvalue. I denote it with p(A). The ...
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37 views

Fixed Matrices over finite field by a map

Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. Let us consider a map $f:M_n$ $\longrightarrow$ ...
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27 views

Tensor tranformation between basis?

If I am the basis vector $e_i$ into another basis to get $e'_j$ I use: $$e'_j=S_{ij}e_i$$ My text book says that $S_{ij}$ is the ith component of the vector $e'_j$ with respect to the unprimed basis. ...
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57 views

Basic Eigenvalue Question

The rotation matrix $$T=\left[\begin{array}{c c}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right]$$ has no eigenvectors as an operator $T:\mathbb{R}^2\to\mathbb{R}^2$. Here ...
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28 views

Orthogonality and linear independence about polynomial vectors

How to prove that polynomial vectors $\left \{ 1,x^{1},x^{2},...,x^{n} \right \}$ are linear independent, but not orthogonal?
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27 views

Find the triangular matrix and determinant.

I have a 4x4 matrix and I want to find the triangular matrix (lower half entries are zero). $$A= \begin{bmatrix} 2 & -8 & 6 & 8\\ 3 & -9 & 5 & 10\\ -3 & 0 & 1 & ...
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3answers
59 views

Determining if vector space or not.

I am having a lot of trouble with what a vector space is and how I would determine if something is a vector space or not. The question I have to answer is: Let $S$ be the set of all vectors ...
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1answer
109 views

standard or orthogonal basis

Standard basis vectors for R^3 are ( 1 0 0 ) , ( 0 1 0 ) and ( 0 0 1 ) . //(All Vectors are of order 3-by-1. If we want to insert vector u into this Basis, then which vector from Standard basis can ...
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67 views

Matrix with all eigenvalues $0$ but not triangular?

Is the situation described in the title achievable? I am looking for a $3\times 3$ case specifically.
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35 views

Extension to the complex numbers for ex. 12 in ch. 6 of Axler's “Linear Algebra Done Right”

I'm wondering how the answer to Sheldon Axler's exercise 12 of chapter 6 "Linear Algebra Done Right" changes when the underlying field is extended from the reals to the complex numbers. The exercise ...
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1answer
29 views

Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with. Can I claim ...
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33 views

Composition of injective linear maps.

I was looking at some solutions for my homework and I didn't understand this part: $S_1,\ldots,S_n$ are injective linear maps such that $S_1S_2 \dots S_n$ makes sense. Prove $S_1S_2 \dots S_n$ is ...
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1answer
48 views

Are isometries always linear?

Let $E$ be a finite dimensional vector space (over a field of characteristic zero) and $f : E \rightarrow E$ be an isometry fixing 0. Must $f$ be linear in this case ? Note : I am NOT assuming that ...
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Existence of Linear Maps and the Fundamental Theorem of Linear Maps.

Prove that there does not exist a linear map $T: \Bbb R^5 \to \Bbb R^5$ such that $\operatorname{range}(T) = \operatorname{null} (T)$. My proof goes like this: Suppose for the sake of contradiction ...
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2answers
51 views

invertibility of self adjoint operators

prove that if $T$ is a self adjoint operator and $a^2$ is less than $4b$ Then $T^2$+$aT$+$bI$ is invertible. Where $a$ and $b$ are scalars and $I$ is the identity operator Not: please dont use ...
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34 views

most general form of $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution $X = f(A,B,C)$?

What is the most general form of the cubic matrix equation $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution of the form $X = f(A,B,C)$, where $A,B$ and $C$ are positive definite ...
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1answer
44 views

Positive definite martix

I understand the majority of this solution, it's just I don't understand why I have to use both $\epsilon_1 $ and $\epsilon_2 $ rather than just $\epsilon$. I understand that i'm working with ...
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19 views

Power iteration sequemce for a special nonnegative irreducible imprimitive matrix

Let $A \in \mathbb{R}^{n \times n}$ be nonnegative irreducible matrix with maximum positive eigenvalue equal to 1. Let's assume $A$ has $h$, $h > 1$ eigenvalues $\lambda_1, \dots, \lambda_h$ with ...
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3answers
68 views

Determinants Proof

Let A and B be square matrices. Prove (or disprove) the following $$\det(qA) = q^{n} \det(A).$$ I tried disproving it with counterexamples but I could not find one.
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find an ellipsoid given its intersection with axes and knowing the lengths of its principal axes

My question is about ellipsoids. I have an ellipsoid in 3D centered at zero so it has an equation: $x^T U \Sigma^2 U^T x = 1$ I know the lengths of it's principal axes (therefore the $\Sigma$ ...
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3answers
41 views

Question about eigenvalue and determinant

Question : If $A$ is an $n \times n$ matrix with real entries and $n$ eignvalues $(a_{1},a_{2},a_{3},a_{4},\cdots,a_{n})$, then does $det|A|=a_{1}a_{2}a_{3}a_{4}\cdots a_{n}$? (THE ANSWER IS : YES) ...
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1answer
39 views

Determinant of Identity minus a singular matrix

I am interested in calculating, or bounding in some way, the following determinant \begin{equation} \det\left[\mathcal{I}-Rxx^t\right] \end{equation} Here, $Rxx^t$ is clearly a singular matrix. Im ...
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0answers
39 views

diagonal of pseudoinverse of laplacian matrix

I have to find the diagonal of the pseudoinverse of a laplacian matrix evaluated on a directed and weighted graph. My laplacian is defined as: L = D - A where: D is a diagonal matrix; Di,i the sum ...
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1answer
28 views

How the Wronskian works

To prove linear independence of a set of functions, we say that given their Wronskian matrix W, Wx = 0 implies trivial solution (0,0,0,...) if the value (determinant) of the Wronskian is identically ...
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1answer
22 views

Linear Algebra - elimination and linear systems

By given this matrix: \begin{pmatrix}1&1&1&0\\2&3&k&1\\3&k&5&1\end{pmatrix} I need to find, what are the values of k the system has infinity/single/no solution. So ...
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1answer
34 views

Prove that the elements of these two sequences are not null

Let $x_{n+1}=x_n+2y_n$ and $y_{n+1}=y_n-x_n$, where $x_1=1$ and $y_1=-1$. I tried proving by contradiction, I tried by induction, I got nothing. This is a question I had on an exam, I didn't manage ...
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If $A^2 = B^2$, then $A=B$ or $A=-B$

Let $A_{n\times n},B_{n\times n}$ be square matrices with $n \geq 2$. If $A^2 = B^2$, then $A=B$ or $A=-B$. This is wrong but I don't see why. Do you have any counterexample?
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why is algebraic multiplicity always equal to the geometric multiplicity of distinct eigen values corresponding to Symmetric matrices?

In other words why is symmetric matrix always diagonalizable? could someone explain intuitively?
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80 views

Caracterization of isometries that preserve time-orientation in $\Bbb L^3$

First of all, I'm considering $\Bbb L^3$ with the convention: $$\langle (x_1,y_1,z_1),(x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ Let $\Lambda = (\lambda_{ij})$ be an isometry of $\Bbb L^3$. I ...
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2answers
32 views

Matrix Algebra, Signs of solution

I have a system $AX = B$, where $A$, $B$ and $X$ are $N \times N$ matrices. I am interested in the properties of the solution $X$. $B$ has the following property: the diagonal terms are strictly ...
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25 views

Find the angle of rotation about a vector caused by application of a rotation matrix

I have a rotation matrix $R$ and a unit vector $\mathbf{v}$. How can I find the angle of rotation about $\mathbf{v}$ caused by the application of $R$?
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Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
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A question in matrix theory, SVD related.

For four $m\times n$ matrices A, B, A', B'. If $AA^\dagger=A'A'^\dagger, BB^\dagger=B'B'^\dagger$ and $AB^\dagger=A'B'^\dagger$, then if there always exists an unitary matrix V in U(n) such that ...
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2answers
35 views

Least Squares Solution Confusion

Say if I have an overdetermined system $A\vec x=\vec b$, I can use the normal equations $\implies$ $A^TA\vec x=A^T\vec b$. If I solve for $\vec x$ I will get a "solution" with an error. It says in ...
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2answers
35 views

Proving a subset is a subspace of a Vector Space

To prove a subset is a subspace of a vector space we have to prove that the same operations (closed under vector addition and closed under scalar multiplication) on the Vector space apply to the ...
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1answer
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Incremental Cartensian Coordinates Betwwen Two Known Coordinates

I've done a lot of searches and haven't found exactly what I'm looking for. I'm looking for an algorithm that will provide me the cartesian coordinates (xyz) every 100ft between two known cartesian ...
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1answer
86 views

Relationship between eigenvectors of matrices

I am investigating parameter estimation in reduced-rank regression and have come across the following linear algebra result which I haven`t been able to prove. Suppose, $A \in \mathbb{R}^{nxm}$ of ...
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1answer
28 views

Minimze min max (A*x)

has this example matrix A some special propertries, which might be useful? $$ \left[\begin{array}{rrrrrr} 3 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 4 & 3 & 0 & ...
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3answers
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Proof needed for this exercise from “Linear Algebra Done Right”

Suppose that $U$ and $V$ are finite-dimensional vector spaces and that $S\in \mathcal{L}(V,W)$ and $T\in \mathcal{L}(U,V)$, where $\mathcal{L}(X,Y)$ is the vector space of linear transformations from ...
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basis exchange linear equation

If you have: $Ax=b$ If you apply a basis change to $A$ and to $b$. Is then the solution(s) $x$ the same? If $A$ is a sparse Matrix is then $A$ still a sparse Matrix?
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2answers
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Differentiation of polynomial as a linear map

Define D: P$_{2}$($\mathbb{R}$) $\mapsto$P$_{2}$($\mathbb{R})$ by D(p)(x) = p'(x) , Show D is linear? . Im a little unsure why this relation is linear?, if we let p(x) = ax$^{2}$+bx+c then p'(x) = ...
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40 views

How to determine for which value of an unknown parameter, one eigenvalue is 0?

Given the matrix: $A = \begin{bmatrix} a & 1 & 0 \\ 4 & a & 1 \\ 0 & 0 & a \end{bmatrix}$ for which value of the parameter $a$ one eigenvalue is certainly equal to $0$? ...
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4answers
550 views

When does $Ax=b$ have any solutions?

Suppose $A$ is a $3\times 3$ matrix with columns $v_{1}, v_{2}, v_{3}$. If $b = 2v_{1}- v_{3}$, then $Ax = b$ has one or more solutions. Is this true or false? Is this false, since $Ax= b$ has ...
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1answer
41 views

Calculate Points Between Two Points

I have two points $(A, B)$, both with longitude and latitude. For each point I have a speed in $km/h$, I assume a car drives from point $A$ to point $B$. I already have a function to calculate the ...