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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Multiple Cartan sub-algebras

How is it that for a Semi-simple Lie Algebra there is not one Cartan Sub-Algebra? I assume since there are multiple CSA's of a SS Lie algebra that must mean some of the ss elements of the Lie ...
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2answers
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No Solution Set (A|b)

Let there be $Ax=b$ a set of m equations with n unknowns. If $\operatorname{rank}(A) \neq \operatorname{rank}[A|b] $ the system of equations does not have a solution Does it mean that: there is a ...
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Subspace in 2D Help

I understand that any line through the origin is a subspace of a vector space. Why won't a curved line through the origin be a subspace in 2D? When I think of a line through the origin, I think of a ...
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Basis and Linear Transformation Proving Problems [on hold]

Can someone help me with these problems? I'm really poor at proving.
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Diagonalizabilty of ad(adjoint map)?

let $\mathsf{g}$ be a finite dimensional lie algebra and $\xi\in\mathsf{g}$. Under which conditions the adjoint map $ad_\xi :\mathsf{g}\longrightarrow \mathsf{g}$ is diagonalizable? what about ...
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275 views

Powers of Irreducible Transition and Periodic Transition Matrices

Suppose P is irreducible transition matrix with period d. How many communicating classes does P^k have and what is the period of each state?
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Sum of Determinants = Scalar product with normal vector?

Today I have seemingly simple question and maybe someone knows the answer without getting into messy calculations. So we have $n$ vectors $v_1,\dots,v_n\in\mathbb{R}^n$ and let us assume for the ...
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1answer
212 views

general formula for an orthogonal projection of a point onto a line

Could someone confirm this or correct the mistakes because this seems somehow wrong although I doublechecked it. Mx,My are coordinates of a point and Px,Py,Kx,Ky are coordinates of a line on a ...
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1answer
39 views

A solution of a linear system in some extension field implies a solution in the subfield

Fix a field extension $k\subseteq K$ (assume that we are in characteristic $0$) and consider a linear system $Ax=b$ where $A$ is a matrix (not necessarily square) with coefficients in $k$. I don't ...
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2answers
97 views

nth derivative of determinant wrt matrix

I'm working on an expression for the nth derivative of a (symmetric) matrix, i.e. \begin{equation}\frac{\partial^{n} \det(A)}{\partial A^{n}}\end{equation} Starting with \begin{equation}\frac{\partial ...
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0answers
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Matrix inequalities question

Let $A, B \in \mathbb{R}^{n \times n}$. Assume that: $$ 0 \preccurlyeq 2 A^\top A \preccurlyeq A^\top + A $$ $$ B^\top + B \preccurlyeq 0 $$ Is the following inequality true? $$ A B + B^\top ...
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Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
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An interesting linear algebra question

Let $A$ and $u$ be $n\times n$ matrix and $n\times 1$ vector of $\mathbb{C}$. Denote $\overline{A}$ is the matrix $(\overline{A})_{ij}=A_{ij}^*$, the conjugate number; ($\overline{A}$ is not the ...
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5answers
50 views

How to find the point of intersection with three equations?

Given the following equations with three variables $a, b, c$ $a-5b+4c=-3$ $2a-7b+3c=-2$ $-2a+b+7c=-1$ How can I determine the point (if it exists) at which all three lines intersect?
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0answers
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what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
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2answers
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dimension of vector space such that $MN=0$

Consider the matrix $\displaystyle M=\begin{bmatrix} 1 &0 &-1 \\ 0 &1 &0 \\ 1 &1 &-1 \end{bmatrix}$ and let $S_M$ be the set of $3\times3$ matrices N such that $MN=0$ . ...
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2answers
28 views

Prove that function is inner product

$V$ is a space of polynomials, we have $p=a_0+a_1x+\dots +a_nx^n$ og $q=b_0+b_1x+\dots +b_nx^n$. I need to show this function is an inner product: $$\langle p,q\rangle=\sum_{j=0}^n ...
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2answers
268 views

Orthogonal complement of the diagonal matrices in the inner product space of matrices

$V$ is the matrices space (scalar over the complex). definition of inner product space is: $(A,B)=tr(AB^*)$. $A$,$B$ matrices. assuming $D$ is the subspace of all Diagonal matrices. I need to find ...
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1answer
15 views

Canonical form for orthogonal similarity classes

Could someone point me to a reference re canonical forms for classes of matrices in $M_n(\mathbb{C})$ which are unitarily similar? That is, canonical representatives for the equivalence class defined ...
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2answers
47 views

An elliptic element of $SL(2,\mathbb{R})$ conjugate to a rotation

Show that an elliptic element of $SL(2,\mathbb{R})$ is conjugate to a rotation. An element $A$ of $SL(2,\mathbb{R})$ is called an elliptic element if $|\text{tr}(A)|<2$ As $|\text{tr}(A)|<2$ ...
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2answers
24 views

Need verification - Given a Hermitian matrix and two eigenvectors corresponding to distinct eigenvalues, show x and y are orthogonal.

Claim: Let $A \in \mathbb{C}^{mxm}$ be hermitian ($A = A^*)$. If $x$ and $y$ are eigenvectors corresponding to distinct eigenvalues, then x and y are orthogonal. Proof: Let $x$ and $y$ correspond to ...
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1answer
38 views

The unit vector in the direction of u

I'm trying to work my way through a linear algebra assignment, and I'm struggling with a few questions. This is one of them. I'm completely lost at question A. I think that B's answer is true, but ...
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1answer
53 views

Finding the smallest max eigenvalues for related matrices?

While messing around with a spectral approach to a graph coloring question, I happened upon a type of problem I hadn't seen before. Suppose you have two symmetric $n$ x $n$ matrices in the form ...
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Kronecker product and the vec operator: confusion on proof of vec(AXB) = (transpose(B) ⊗ A) vec(X)

I was reading up on Kronecker products and vec operator from couple of sources and landed on the equation: vec(AXB) = (transpose(B) ⊗ A) vec(X) suppose ...
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1answer
259 views

prove that $T$-cyclic subspace of $V$ generated by $x$ is $T$-invariant

Let $T$ be a linear operator on a vector space $V$, and let $x$ be a non-zero vector in $V$. The subspace, $$W = \operatorname{span}(\{x,T(x),T^2(x),\ldots\})$$ I have to prove that $W$ is a ...
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1answer
30 views

If $T$ is not diagonalizable over $\Bbb R$ and $T$ has all its eigenvalues real, then can I say $T$ is not normal?

$\Bbb V$ is a vector space of dimension $n$ and $T$ is a linear operator on $\Bbb V$ I know that if $T$ is not diagonalizable over $\Bbb C$ then $T$ is not normal. My question is if $T$ is not ...
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1answer
26 views

Don't understand Levi decomposition theorem

Levi decomposition theorem states that any finite-dimensional real Lie algebra $L$ is the semidirect product of a solvable ideal and a semisimple subalgebra. I don't understand this since to me it ...
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2answers
27 views

Show that every row of matrix $S$ is a linear combination of its bottom row and the row (1 1 1 1 1 1 )

Couldn't solve the following three questions. $$S=\begin{pmatrix} 36 & 35 & 34 &33&32&31 \\ 25 & 26 & 27&28&29&30 \\ ...
4
votes
2answers
157 views

Show A and B have a common eigenvalue

Let A, B and C complex square matrices such that: $ C\neq 0 $ and $AC=CB $ prove that A and B has a common eigenvalue. It's worth mentioning that earlier in the assignment I have proved that ...
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3answers
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Finding the order of elements in a Galois Field

Does there exist a Galois field GF(4)? GF(4)={0,1,2,3}; If we take this Galois field, then the element '2' is not having any degree..? So is it possible to construct GF(4) ?
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Conjugating by an upper triangular matrix does not change the diagonal entries.

This post http://mathoverflow.net/questions/49679/a-matrix-similarity-problem makes the claim that conjugating by an upper triangular matrix does not change the diagonal entries. But how do I prove ...
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2answers
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Midpoint of chord of contact

Question: The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is: a) $20(x^2 - y^2)- 36x + 45y = ...
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1answer
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Reentrant constraints in active set algorithm?

Problem definition Supposing you're trying to solve a quadratic program: $$ \min_x f(x) = \frac{1}{2}x^T Q x + c^T x \\ \mbox{s.t} \, \; A x \ge b$$ Where Q is square ($n$x$n$), positive semi ...
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2answers
39 views

finding the dimension of a matrix, the sum of whose rows is zero

Let $V$ be a vector space of $n\times n$ matrices over $R$ and Let $W$ be subspaceof matrices with entries in each row adding upto zero.then the dimension is? n $\frac{n(n-1)}{2}$ $n(n-1)$ $n-1$ ...
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0answers
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Would this thinking about the dot product hold?

Background today I completed the chapter on the dot product of vectors. But in trying to figure out exactly what the dot product is. I came to the conclusion that it can be interpreted as the length ...
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1answer
145 views

How many matrices with integer eigenvalues are there?

Let m,n be natural numbers. How many mxm-matrices with integer entries from -n to n have the property that all eigenvalues (possibly multiple) are integers ? The following table calculated with PARI ...
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2answers
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Continuity in finding eigenvectors

I'm wondering whether there's "continuity" in the eigen vectors of different matrices corresponding to appropriate eigenvalues. For instance, if we change certain elements in a matrix, can we ...
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0answers
23 views

3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix

This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ...
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1answer
88 views

Inverse matrix and its zero entries

Let $A$ be an $N \times N$ square invertible matrix with inverse $A^{-1}$. Is it possible to know through information of $A$ alone (i.e. without actually calculating $A^{-1}$) Which entries of ...
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0answers
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Algorithms for solving overdetermined, homogeneous linear systems with multivariate polynomial coefficients

I would like to solve overdetermined, homogeneous linear systems of equations with multivariate polynomial coefficients, i.e., $Ap=0$ with $A$ an $m\times n$ matrix, $m\gg n$, and $a_{i,j} \in ...
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2answers
25 views

Choose $h$ and $k$ such that the system has, no solution, a unique solution, and many solutions.

Looking through my textbook, I see no examples as to how to solve this \begin{align} x - 3y & = 1 \\ 2x + hy & = k \end{align}
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1answer
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Finding minimum point of a function using linear algebra

Given a function $$q(x,y)=2x^2-2xy +2y^2$$. Find the minimum point of the following function by first converting it to a matrix form and using the diagonalisation of the matrix to find its minimum ...
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3answers
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Span of columns (or rows) of a given matrix?

Consider the following matrix: $$A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ \end{pmatrix}$$ The columns of $A$ span $\mathbb{R}^2$. The columns of $A$ span $\mathbb{R}^3$ ...
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1answer
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Power of a matrix, given its jordan form

Can someone please explain how to find the power of a matrix $A$, given $A=MJM^{-1}$ where the matrix $J$ is in the Jordan canonical form? Or else please explain how to find the powers of a matrix ...
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2answers
40 views

Question about inner products

Given a real or complex vector space $\;V\;$ and a (finite) basis $\;B\;$ of it, does it always exist an inner product on $\;V\;$ s.t. $\;B\;$ is an orthonormal basis with respect to it? The question ...
4
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1answer
74 views

Angle between two planes in four dimensions

Suppose I have two planes defined in 4D space, either in terms of vectors spanning the planes, $X = t_1 A_1 + t_2 B_2$ and $X = t_3 A_3 + t_4 B_4$ (where $X$, $A$'s, and $B$'s are vectors with four ...
197
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11answers
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What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
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3answers
53 views

System of linear equation

Determine the value for k for which the system of linear equation has infinitely many solution. \begin{cases} 2x - y = 2\\ 4x + ky = 4 \end{cases}
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1answer
54 views

Minimum eigenvalue of product of two matrices

Abstract description: Let $\mathbf{A}$ and $\mathbf{B}$ be two $n \times n$ real matrices. Let $\sigma( \mathbf{A B} )$ denote the spectrum of $\mathbf{A B}$. Assume that (A1) $\mathbf{A}$ is ...
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Group inverse of positive semi definite matrix

Group inverse and Moore Penrose inverse of a positive semidefinite matrix are same. How?