Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Orthogonal Complex Matrices

Consider $M_{2\times2}(\Bbb C)$ together with the inner product $<A,B>=Trace(B^\dagger A)$, where $B^\dagger$ is conjugated. Let $W$ be the subspace defined by $W= \left ...
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1answer
96 views

Proof Verification of Cayley Hamilton

I am wondering about how this proof I am doing for the cayley hamilton is and if it is fully valid. I am also interested in any suggestions, better options or things I should note. Or if I am on right ...
2
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3answers
90 views

Prove there are no other invariant subspaces

Let $f \in End(V)$ has $n\times n$ matrix at basis $v_1, … ,v_n$ which is jordan block($n \times n$) ...
2
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0answers
121 views

Properties of eigenvectors of a sample covariance matrix?

My apology if the question is not appropriate. For me Eigenvectors are quite a mystery. Does it have any property that we can relate to the matrix it came from? By property I mean something like the ...
2
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1answer
61 views

Why the principal components correspond to the eigenvalues?

Suppose ${\bf{X}} = ({X_1},{X_2},\ldots,{X_n})$ are the original components (also random variables) and ${{\bf{w}}_j} = ({\omega _1},{\omega _2},\ldots,{\omega _n})$ are loadings for the $j$th ...
3
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2answers
98 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
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2answers
56 views

It is possible to orthogonalize a set of linearly independent vectors via SVD?

Let's say I have a set of linearly independent vectors, collected in a square matrix $\mathbf{M}$. I know that I could orthogonalize these vectors with the QR decomposition, $\mathbf{M} = ...
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1answer
54 views

Basis of a basis-linear algebra?

Usually when we say that $v_1$ and $v_2$ are basis we imply that they are linearly independent and span the space. We by default denote $v_1$ and $v_2$ in $i$-$j$ basis. Then how is $i$-$j$ basis ...
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1answer
47 views

Find linear transformation such that N(T)=R(T)

Well the title quite says it all, only I must say $T:\mathbb{R}^{2} \mapsto \mathbb{R}^{2}$. I understand that $N(T)=\{v:T(v)=0\}=\{T(v):v\in \mathbb{R}^{2}\}=R(T)$ implies that $T(T(v))=0$ for any $v ...
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0answers
84 views

Find the closest element in a span of matrices?

Given a span: $$\left\{\begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}, \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix}\right\}$$ find the closest element to ...
0
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1answer
66 views

Finding basis for eigenspace when RREF returns several non-zero rows

Given a matrix A: $ \left(\begin{array}{rrr} 1 & 1 & 3\\ 1 & 3 & 1\\ 3 & 1 & 1 \end{array}\right). $ The eigenvalues are 5, 2 and -2. Now I have trouble with the eigenvalue ...
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4answers
84 views

Proving $ax+b$ is a linear function

$L\colon\mathbf{R}\to \mathbf{R}$ be given by $L(x)=ax+b$ over the scalar field $R$. I understand for that a function to be linear, it must adhere to the properties of additivity and scalar ...
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1answer
40 views

Node Game Recursion Problem

http://i.imgur.com/LwNr4rn.png I'm trying to figure out part a. However, I'm not sure if the set of simultaneous equations I've found is correct. Or at least, I can't solve the set. Any help would be ...
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1answer
43 views

Showing that following is not a vector space?

I have the following 8 axioms for a Vector Space and the following question. I managed to prove that Axiom 3 doesn't work(and as a result Axiom 4 because 0 element doesn't exist) but the answer ...
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0answers
114 views

Expectation of inverse of a symmetric matrix with gaussian elements

Is there any way to calculate: \begin{equation} \mathbb{E} \; ( H^{T}H )^{-1} \end{equation} assuming that the entries of the matrix $H$ are gaussian random variables with unknown means but same ...
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0answers
104 views

Using Cholesky factorization to solve the system AXA=B

I have been given a problem of solving X, which is an unblurred image, in the system: $$B = A X A \iff X = A^{-1} B A^{-1}$$ Where the matrix A describes the blurring of an image and the matrix B is ...
2
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0answers
45 views

Nearest non-negative solution for $Av=b$

Let $A$ be a $n\times m$ matrix. Let us define the system $$Av=b$$ $$v\geq 0$$ I want to find a solution $v$ of this system that is the closest (euclidean norm) to $v_0$, a given $n$-dimensional ...
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1answer
24 views

Why is there no possibility of no solution in homogeneous systems

Sorry for a newbie question, but can u please tell me, why there's no chance for such thing? For example: 5 | 0 is a homogeneous system, because of the 0 at the end. and 5 =/= 0, so... ...
2
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1answer
48 views

Proving a property of $n$ by $n$ matrices

I'm taking a course on linear algebra and have been asked to prove the following theorem. If we let $A$ be an $n \times n$ matrix over a field $\mathbb{F}$, then there exist scalars $a_1, ..., ...
4
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2answers
59 views

Cycle structure of affine transformation

Consider the ring $\mathbb{Z}_n$ of remainders modulo $n$ for some number $n.$ Let $a,b \in \mathbb{Z}_n$ and consider the map $$f_{a,b}(x) = ax+b.$$ If $a$ is invertible then the above map is ...
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0answers
15 views

linear algebra : to find relevant structure for 3 stage interaction

A family has 10 members, of which all cannot interact with each other at all times. Obtain all possible ways in which their interaction (between two persons) can take place. Assuming that the ...
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2answers
28 views

Transformation determined by a basis is linear

Suppose I have a finite-dimensional vector space $V$, and suppose that $(v_1,\ldots,v_n)$ is a basis for $V$. If I define $T:V\to W$, which $W$ is a vector space over the same field as $V$, and for ...
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0answers
66 views

Transformation that f(A+B)=f(A)+f(B) and f(AB) = f(A)f(B)

if $f:M_{n*n}(F) \rightarrow M_{m*m}(F) $ and f transformation identity to identity matrix and $f(AB) = f(A)f(B) , f(A+B)=f(A)+f(B)$. now we want to prove there is an integer like $k$ that $m = k*n$
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2answers
457 views

degree of nilpotent matrix

I need to prove or disprove that A nilpotent matrix's degree is less than or equal to its dimension. I tried to make a counterexample but I found nothing and I think that this claim is true but I ...
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1answer
65 views

every real symmetric matrix has at least one real eigenvalue. [duplicate]

Why every real symmetric matrix has at least one real eigenvalue?
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2answers
208 views

Write down a homogeneous linear system of three distinct equations in three variables that has the non-trivial solution $(x, y,z) = (1,2, 4)$

Write down a homogeneous linear system of three distinct equations in three variables that has the non-trivial solution $(x, y, z) = (1,2, 4)$. I am confused on how to approach this problem
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1answer
58 views

Proving if $(A-B)^2 = (A+B)^2$, then $A^2B = BA^2$

When you multiply out $(A-B)^2 = (A+B)^2$, I get $-AB=BA$. I then multiply by -$A$ to get $A^2B = -ABA$. I don't see how I'm supposed to get $BA^2$.
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1answer
79 views

Diagonalization: How to show that A exists $S^2 = D$ given that D is a nonegative diagonal marix

(a) Show that if D is a diagonal matrix with nonnegative entries on the main diagonal, then there is a matrix S such that $S^2 =D$ SOLVED (b) Show that if A is a diagonalizable matrix with ...
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3answers
65 views

How do you prove (or disprove) the statement: If $A^3 = 0$, then A-I is non-singular

I've proven something similar: A*A =0, then A + I is non-singular for 2x2 matrices. But not sure how to proceed for $A^3 = 0$, then A-I is non-singular Also, not sure how to prove A*A =0, then A + I ...
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0answers
154 views

Adjoint of Exponential Map

If $\exp: T_p(G) \rightarrow G$ is the expoenential map of a lie group, then what does the adjoint operator (as in $\langle Ax,y\rangle=\langle x,A^*,y\rangle$) of the derivative of exp look like? ...
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2answers
113 views

Better Gaussian Elimination for solving $Ax=b$ [closed]

We know that Gaussian Elimination is very popular method to resolve $Ax=b$. Does anyone know better method than Gaussian Elimination in term of time complexity? Second question,if I assume that A is ...
0
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2answers
92 views

If $A^3 = A$ then the eigen values are all 1 right?

Since $A^n = PD^nP^{-1}$ where D is a matrix consisting only of the eigenvalues of on its leading diagonal. For the scenario to be true $D^B = D$ which is only true if the eigenvalues are all 1s ...
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2answers
58 views

Diagonalization: Am i finding these eigenvectors wrongly?

$$A=\begin{bmatrix} 1&-2&-8\\ 0&-1&0\\ 0&0&-1 \end{bmatrix}$$ $$P=\begin{bmatrix} 1&-4&1\\ 1&0&0\\ 0&1&0 \end{bmatrix}$$ Confirm that P diagonalizes A. ...
0
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1answer
35 views

Quadratic graph / standard form

If I draw a graph of the quadratic $x^2-9=0$, I have a parabola with roots $x=3$ and $x=-3$ and a vertex of $(0,-9)$ with the parabola opening upwards as $a$ is positive in the standard quadratic ...
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2answers
45 views

What does component being zero in particular dimension mean?

I have asked a question on stack physics , basically asking why is it so that every $4\times 1$ matrix can't be written as tensor product of two $2\times1$ matrices ? (for more detail : 4-D column as ...
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2answers
119 views

General Information about Eigenvalues for an 3x3 symmetric matrix

How do we find general information about the eigenvalues of an arbitrary 3x3 symmetric matrix without resorting to explicitly computing the solutions to the cubic characteristic equation?: ...
9
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1answer
405 views

Here is a riddle that I have no idea how to solve.

Okay, so I was trying to solve this riddle found here. It is a diagram of a star with 16 points. Each point corresponds uniquely to a number between 1 and 16. The letters on each point represent a ...
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0answers
86 views

Dwell times of an absorbing markov chain conditional on reaching specific absorbing state

The fundamental matrix of a discrete time markov chain with absorbing states dictates the expected amount of time spent in each state $j$, given that you started in state $i$. The equation is $$S = ...
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1answer
26 views

prove or disproof problem about linear transformation

prove or disprove: if $V$ is a vector space and $T:V\to V$ is a linear transformation so that $\mathbf T^2= 0$ (the zero transformation) then $\operatorname{Im}(T)\subseteq\ker(T)$ have no clue...
2
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1answer
82 views

Gram-Schmidt Process.

I have this question on as problem set. Find an orthonormal basis for M2x2(R) by applying the Gram-Schmidt process to the basis. ...
1
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1answer
75 views

Left eigenvector of stochastic matrices with eigenvalue 1

I am only talking about matrices for finite number of states. By the existence of unique equilibrium distribution, this surely means there can only be one of such eigenvector (i.e. the eigenvalue 1 ...
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1answer
40 views

How to solve this matrix determinant?

I can't solve this problem, I know that is too easy but I don't how to. Show that $$ \det\begin{bmatrix} a+x & b+x & c+x \\ b+x & c+x & a+x \\ c+x & a+x & b+x \end{bmatrix} = ...
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2answers
159 views

What is the derivative of a matrix w.r.t itself?

what is the derivative of \begin{equation}\partial \frac{x^TVx}{\partial V} \end{equation} where V is a matrix and x is a vector. In general what is the right way to calculate matrix derivatives w.r.t ...
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0answers
31 views

getting confused with parameters in linear algebra's problem

Ok, so I got this problem: 4x+8y+7z+3cw=3b x+2y+2z+cw=b 2x+4y+2z+(c-1)w=b and I'm doin' the first steps as follow: ...
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1answer
119 views

A curious way to write the eigenvectors of the Boolean hypercube

It seems that one can write the eigenvectors of the hypercube $\{ \pm 1\}^n$ as the functions, $\{ \chi_S \}_{S \subseteq [n] }$. And these functions $\chi_S$ are defined on the vertices $x \in \{ \pm ...
2
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0answers
55 views

det equation over the integers

Let $ A\in {M_2 (Z)} $ . If $\det (A^3+I)=1$ -find $A$. I tried using eigenvalues and the fact that if they are not rational (or if they are not real) - they must be conjugated but the calculation is ...
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2answers
102 views

Intuition of vector of $n$ dimensions.

Actually I was reading Lectures of Physics by Feynman where he introuduced gradient theorem. While I was googling about this theorem, I came across vector of $n$ dimensions. I'm having a great problem ...
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2answers
41 views

Dimension of a subspace smaller than dimension of intersection

Suppose I have a finite-dimensional vector space $V$, and $U_1, U_2$ are subspaces of $V$, such that $U_1\nsubseteq U_2$. Is it possible that $\dim{U_1}\leq\dim{U_1 \cap U_2}$?
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1answer
38 views

Question about determinants 2

I understand the first part, but I'm a bit confused on how to go about the second part. I get to $$t_1^2-\operatorname{Tr}(A)t_1=t_2^2-\operatorname{Tr}(A)t_2=-\det(A)$$ but don't know how to ...
3
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1answer
41 views

What is the base and dim for the kernel of this linear transformation version2

Ok, so i have a linear transformation that is from second degree polynomial to a $2\times 2$ matrix $$T : \mathbf{P_{2}[X]} \to \mathbb{R^{2\times2}}$$ which defined as: $$T(P(X)) = \begin{pmatrix} ...