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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Is there a great book on eigenvalues?

I keep encountering ostensibly very different branches of mathematics, only to have eigenvalues show up in each one. Is there a single book out there that presents a deep, unified account of the ...
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Quantum Information: Deutsch-Jozsa Algorithm

There is a step in the construction of this algorithm which I'm not understanding: $\displaystyle \left[\sum_x \frac{| x \rangle}{\sqrt{2^n}}\right]\left[\frac{ | 0 \rangle -| 1 \rangle ...
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Is the identity matrix an elementary matrix?

My reasoning is yes, as you can switch row i with row i in the matrix... But I'm not sure if it's a "legal" elementary operation to switch a row with itself.
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Is there any square matrix with orthonormal rows but not orthonormal columns?

An orthogonal matrix necessarily has orthonormal columns, and orthonormal columns necessarily give an orthogonal matrix. Also, orthonormal columns imply orthonormal rows. But how about the converse ...
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Matrix groups generated by translation and inversion in the unit sphere

Let $\alpha$ be algebraic over $\mathbb{Q}$, and consider the subgroup $G$ of $\mathrm{SL}_2(\mathbb{C})$ generated by inversion in the unit sphere and translation by $\alpha$. That is, consider ...
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Relation between Interior Product, Inner Product, Exterior Product, Outer Product..

Following my previous question Relation between cross-product and outer product where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot ...
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How to find matrix form of an operator on a vector space V which is direct sum of its two subspaces?

I am studying a lecture notes where I found this result: Let $A$ be an operator in vector space $V$ . If $V$ is equal to direct sum of its two subspace $U$ and $W$ ie $V = U\bigoplus W$, and if the ...
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Relation between cross-product and outer product

If inner products ($V$) are generalisations of dot products ($ \mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($ \mathbb{R}^3$) in some way? A quick search reveals that ...
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123 views

Lower bound of $J=\frac{x^TAx}{x^TBx}$

Consider two symmetric positive semi-definite matrices $A, B \in \mathbb{R}^{n\times n}$. Suppose that $A$ and $B$ have the same null space $\mathcal{N}\subset \mathbb{R}^n$. Now consider the ...
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Is this a Polygon?

According to the definition of Polygon, If a Poly-line's first and last points are connected then it is called Polygon. See the image below. I have P1, .... P5 Polyline. If I draw a line from P5 to ...
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Intuition in permutations for Laplace Determinant Expansion

Starting with the Leibniz formula for the determinant, I wish to derive the Laplace (Cofactor) Expansion. At the risk of being overly verbose, please see the proof here. Now I understand the idea of ...
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linear transformation $T$ such that $TS = ST$

Let $V$ be a finite-dimensional vector space over $F$. Let $T:V \rightarrow V$ be a linear transformation such that $ST=TS$ for all linear transformations $S:V \rightarrow V$. Show that $T = cI_v$ for ...
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For $A \in \mathbb{R}^{3 \times 3}$, find $P, Q \in \mathbb{R}^{3 \times 3}$ such that $A = P-Q$, where $P^2 = P$, $Q^2 = Q$ and $PQ = 0 = QP$

This is an exercise from a previous linear algebra exam: The diagonalisable matrix $$A = \begin{pmatrix} 3 & -6 & 2\\ 4 & -7 & 2\\ 8 & -12 & 3 \end{pmatrix} \in ...
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516 views

Fast algorithm for LU factorization

If A is a symmetric matrix, is there a fast algorithm for LU factorization? I know this algorithm for non-symmetric matrix. ...
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273 views

A bit of direct sum confusion

So I'm reading through Serre's "Linear Representations of Finite Groups," and I'm a bit confused by what's probably a fairly minor point. However, subsequent proofs are hinging on it, so I figure ...
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345 views

Commutative matrix

If two matrices $A$ and $B$ are commutative then all rules for real numbers $a$ and $b$ apply for the matrices? For example, if $AB=BA$ then: $(A+B)^2=A^2 + 2AB + B^2$ $A^3 - B^3 = ...
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184 views

Vectors-Scalar Triple product

Show that $$(a \times b) \times (c \times d)= [a,b,d]c - [a,b,c]d.$$ I understand $[a,b,d] = a \cdot (b \times d)$ and $[a,b,c] = a \cdot (b \times c)$ but not really sure where else to go with ...
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76 views

Projecting a vector component out of vector $a$

Suppose a vector $a$ is given. What is precisely meant by projecting the component of vector $b$ out of $a$? Does that mean that the resulting vector $a_1$ (obtained by "projecting out the component ...
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Entries of a matrix raised to any real power

Suppose $A=(a_{ij})$ is an $n\times n$ real matrix and define $T(A)=\max\{|a_{ij}|\}$, where the maximum is taken over $1\leq i,j \leq n$. I know how to show that $T(AB)\leq nT(A)T(B)$ for all $A$ ...
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If $V_0$ is the subspace of matrices of the form $C=AB-BA$ for some $A,B$ in a vector space $V$ then $V_0=\{A\in V|\operatorname{Trace} (A)=0\}$

If $V_0$ is the subspace consisting of matrices of the form $C=AB-BA$ for some $A,B$ in a vector space $V$ then $V_0=\{A\in V|\operatorname{Trace}(A)=0\}$. The problem above is one of the past ...
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T-invariant space $W$ and its complement $W'$

Full version of the problem is following: Let T be a linear transformation on a finite dimensional vector space $V$ over a field $\mathbb{F}$. If the minimal polynomial $p_t$ of T is irreducible, ...
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Quantum Information: Tensor with Outer Product

If $\mid\phi\rangle=a_0\mid 00\rangle + a_1\mid 01\rangle +a_2\mid 10\rangle +a_3\mid 11\rangle$ and $P_0=\mid 0\rangle\langle 0\mid \otimes I$, how do we show that $\langle\phi \mid P_0\mid ...
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157 views

Questions regarding the quadratic form $q: \bigwedge^2 \mathbb{R}^4 \to \bigwedge^4 \mathbb{R}^4, x \mapsto x \wedge x$

An exercise of my last year's linear algebra class asks as follows: Determine the type of the quadratic form $q: \bigwedge^2 \mathbb{R}^4 \to \bigwedge^4 \mathbb{R}^4, x \mapsto x \wedge x$. ...
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Are there diagonalisable endomorphisms which are not unitarily diagonalisable?

I know that normal endomorphisms are unitarily diagonalisable. Now I'm wondering, are there any diagonalisable endomorphisms which are not unitarily diagonalisable? If so, could you provide an ...
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A basic question related with the positive definite matrix

I have a one doubt related with positive definite matrices. Suppose that we have an arbitrary non zero matrix $A$ . Can we find such matrix $B$ which may depend on $A $such that product $AB$ is ...
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135 views

The stucture of invariant polynomials on matrix

Let field $\mathbb F$ be either $\mathbb R$ or $\mathbb C$ and $M_n(\mathbb F)$ all $n \times n$ matrixes. We denote by $I_n(\mathbb F)$ the space of all functions: $P : M_n(\mathbb F) \rightarrow ...
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Singular-value inequalities

This is my question: Is the following statement true ? Let $H$ be a real or complex Hilbertspace and $R,S:H \to H$ compact operators. For every $n\in\mathbb{N}$ the following inequality holds: ...
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107 views

Choosing initial approximation for computing Moore-Penrose inverse

I am trying to compute the Moore- Penrose inverse of a given $m\times n$ matrix $A$. I did convergence analysis then I came across to the following condition. For convergence of my method: $\max\mid ...
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Matrices with elements that are a distinct set of prime numbers: always invertible?

Inspired by a previous question, given a square non-symmetric matrix whose elements are all prime but distinct from each other, does this guarantee that the matrix is invertible? It's easy to see ...
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Functions that generate “easy” matrices of full rank

While explaining how to invert matrices I once used this ill-fated example $A=\begin{pmatrix} 1&2&3\\4&5&6 \\7&8&9 \end{pmatrix}$ which can not be inverted ($\det(A)=0$). That ...
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Computation of determinant of a matrix with elements from an arbitrary commutative ring

The cofactor formula for computing the determinant of a matrix is applicable when elements of the matrix are from a commutative ring. However, the complexity of this method is extremely high and I ...
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Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
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Find the general matrix commuting with a Jordan canonical Matrix

I need to find the most general matrix X commuting with $ J= D_{g} [J_{2}(2), J_{1}(2), J_{2}(3), J_{1}(3)] $ I also need to find the dimension of $ C(J) $ the centralizer of $ J $. I have found ...
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319 views

Find the Jordan Canonical Form from c(x) and m(x)

Given the matrix B: $$ \begin{pmatrix} 2 & 1 & -2 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ \end{pmatrix} $$ I have found that the ...
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Automorphism groups of real clifford algebras

I'm sure someone has already worked-out what all the relevant groups really are; my question is about how signature duality interacts with these groups. So, by an awful calculation, and choosing a ...
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Same Morse coordinates for different Morse functions

Let $f,g\in C^\infty(\mathbb R^n;\mathbb R)$ be two Morse functions having both a critical point at $0$. Is it always possible to find local coordinates around $0$ such that both $f $ and $g$ become ...
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What is the reflection in linear algebra

as we know projection $A/B = AB^t(AB^t)\cdots B$ how about reflection? do it have orthogonal reflection or oblique reflection? what is the reflection in linear algebra Reflection $= 2(A/B) - A$ ...
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Solving for specific entries in a Lyapunov Equation

Let $A$ be a $2n\times 2n$ real matrix with the following structure \begin{equation} A = \left(\begin{matrix} 0 & -I \\ K & S \end{matrix}\right) \end{equation} with all sub-matrices of size ...
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Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
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Solving linear system of equations when one variable cancels

I have the following linear system of equations with two unknown variables $x$ and $y$. There are two equations and two unknowns. However, when the second equation is solved for $y$ and substituted ...
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Why does the bilinear form vanish between two direct factors?

I extract a problem from the book I am reading: Let $R$ be a field, $A$ be a semisimple split $R$-algebra (associative with $1$). Let $A = \oplus_{n=1}^t A_n$ be a decomposition of $A$ into simple ...
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How can I get matrices for practicing Jordan normal form?

I would like to practice the algorithm for the transformation from a matrix to its jordan normal form (with change of basis). To do so, I wrote this script that generates random $n \times n$ ...
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Prove: symmetric positive definite matrix

I'm studying for my exam of linear algebra.. I want to prove the following corollary: If $A$ is a symmetric positive definite matrix then each entry $a_{ii}> 0$, ie all the elements of the ...
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The slope, i.e. the first derivative transforms covariant, some problems with understanding

I am working through tensoranalysis and the notations of co-/contra- and invariant, guess I understand. If I have a linear transformation, and the coordinates of my vector transform in the opposite ...
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628 views

lagrangian minimisation problem and Karush-Kuhn-Tucker conditions

A rectangular box without a lid is to be made from 50m² of cardboard. Find the maximum volume of such a box.( i know how to solve this in the conventional way, i am trying to figure out how to do it ...
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How to invert a very regular banded Toeplitz matrix?

What's the best way to invert a simple Toeplitz matrix of the following form? $$ A = \begin{bmatrix} 1 & a & 0 & \ldots & \ldots & 0 \\\ a & 1 & a & \ddots & ...
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Is matrix positive semidefinite?

I am new here and have an interesting question: Consider the (n x n) symmetric and real Matrix M with $\sum_j M_{i,j} = 0$ $\sum_i M_{i,j} = 0$ $M_{i,i} > 0$ It seems a matrix of this type ...
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Duality, Symmetry, Dual Spaces

The following is from the book "Tensor Methods in Statistics", which could be downloaded there http://www.stat.uchicago.edu/~pmcc/tensorbook/ I have a question regarding section 0.3.1, titled ...
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327 views

Singular or non-singular matrices

Which of the following matrices are non-singular? $I + A$ where $A$ not equal to $0$ is a skew-symmetric real $n\times n$ matrix, $n\geq 2$. Every skew-symmetric non-zero real $5 \times 5$ matrix. ...
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Can an eigenvalue (of an $n$ by $n$ matrix A) with algebraic multiplicity $n$ have an eigenspace with fewer than $n$ dimensions?

Is it possible for a matrix with characteristic polynomial $(λ−a)^3$ to have an eigenline (one-dimensional eigenspace)? I know that geometric multiplicity can generally be smaller than algebraic ...