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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2answers
48 views

(Linear algebra) if $A$ is normal matrix then, eigenvectors of $ A$ are orthogonal.

I know that the eigenvectors of a unitary matrix are orthogonal. Then is that also true for a normal matrix? How do I prove?
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3answers
30 views

If $V$ is a vector space $\neq$ the vector space of its additive identity alone, must $V$ have a subspace $\neq V$?

It seems to me highly plausible that every vector space $V$ such that $V$ does not consist of the additive identity alone has a (nontrivial) subspace $\neq V$. But I have not yet seen a way to prove ...
0
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0answers
6 views

Parametrization of the split orthogonal group O(n,n)

I would like to find or construct an explicit parametrization of the $2m$-by-$2m$ matrix representation of the real indefinite orthogonal group $O(m,m)$ associated to the bilinear form with matrix ...
0
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1answer
17 views

Inner Product Space and Linear Mapping Theorem

I'm having some trouble proving the following theorem: Let $($$X$,$\langle\cdot | \cdot\rangle$$)$ be an inner product space and $f: X \to \mathbb{R}$ a linear mapping. Prove that there exists a ...
0
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1answer
8 views

“cover the unit sphere by c-fine grid” to prove the vector length preserved by random projection?

The below figure is extracted from the paper http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4031351 . I did not understand the techniques used in the proof, namely, 1."cover the unit ...
3
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2answers
6k views

How can I construct the matrix representing a linear transformation of a 2x2 matrix to its transpose with respect to a given set of bases?

I have been given that I am working with the space of all 2x2 matrices. The basis $B$ for this space is given as a set of four 2x2 matrices, each with an entry of 1 in a unique position and zeroes ...
1
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1answer
606 views

Smith Normal Form

Would the Smith Normal Form of the following matrix over $\mathbb Q[x]$ $$\begin{pmatrix}   (x+a)(x+b) & 0 & 0 &0 \\  0 & (x+c)(x+d) & 0 & 0 \\   0 ...
1
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1answer
27 views

Convergence of the LR algorithm for $2\times 2$ SPD matrices

I've been asked to prove that the following iterations converge to the eigenvalues of SPD $A_0 \in \mathbb{R}^{n \times n}$ $A_0 = \begin{bmatrix}a & b\\ b & c \end{bmatrix}$ with $a \geq ...
0
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0answers
7 views

Which type of correlation should I use?

I am beginner in statistics. I have excel table with few columns. I would like to find correlation between the variables. I have to make an essay to my boss and he wants concrete answers. I searchin ...
-1
votes
1answer
22 views

Question regarding arbitrary parameters

Solve the following system of linear equations: x + y + z = 4 x + y + z = 4 2x + 2y + 2z = 8 I'd like some help understanding how to go about solving this. I ...
0
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1answer
12 views

Is decomposition of a semisimple Lie algebra unique?

A semisimple Lie algebra is defined to be the sum of simple Lie algebras. But is this decomposition to simple Lie algebras unique? If not can you give an example?
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4answers
57 views

If $ax + by = a(b-1) + b(-1)$, then does $x = b-1$ and $y = -1$

In this case, $x$ and $y$ are variables and $a$ and $b$ are arbitrary constants. It seems like just looking at the equation that this would be true, but is there a case when it does not work? If I try ...
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0answers
37 views

Need some hints to solve a problem from “Revue de la filière mathématique”

Let $A\in M_{n}(\Bbb R)$ and $B\in M_{n,m}(\Bbb R)$ and $C=\int_{0}^{1}\exp\left(sA\right)BB^T\exp\left(sA^T\right)\,{\rm d}s$. Prove that $C$ is invertible if and only if $\sum_{i=0}^{n-1} ...
1
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0answers
33 views

squared trace inequality for hermitian matrices

I was wondering how to prove that $Tr(H^2)\cdot d - Tr(H)^2\geq 0$ for each $(d\times d)$ Hermitian matrix $H$. This is equal to $d\sum_j \lambda_j^2-\sum_{j,k}\lambda_j\lambda_k$ with eigenvalues ...
1
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2answers
32 views

Change of basis to find coordinates

This was my previous post: http://math.stackexchange.com/posts/1243265 I changed my question quite drastically and I didn't feel I was asking the question correctly. I attempted with change of ...
0
votes
2answers
15 views

A simultaneous equation question

$38$ bottles of soda was consumed by $18$ women. Some took $2$ and others took $3$ . (A) How many women took $2$ sodas? (B) How many women took $3$ sodas? I thought I might use simultaneous equations ...
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2answers
21 views

Is $Q=V^TWV$ positive definite?

I have the real symmetric $k \times k$ matrix $Q = V^T W V$, where I know $V$ is a $n \times k$ orthogonal matrix (its columns are orthogonal) and $W$ is a $n \times n$ diagonal matrix with all its ...
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1answer
30 views

System of linear equations with four unkowns

I have no idea how to solve this system of equation : $$\begin{align}u+v+w&=7 \\v+w+x&=-8 \\w+x+u&=5 \\x+u+v&=-10\end{align}$$ I usually use the addition/substraction method, but ...
1
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1answer
28 views

Finding the symplectic matrix in Williamson's theorem

tl;dr: How do I construct the symplectic matrix in Williamson's theorem? I am interested in a constructive proof/version of Williamson's theorem in symplectic linear algebra. Maybe I'm just missing ...
0
votes
2answers
23 views

Finding x'y' coordinates from xy coordinates with unit basis vectors

I wasn't really sure where to get started with this question as I don't fully understand what it's asking.. I can see that u1 is made up of i + j (or u2) and that x' is scaled for some scalar k from ...
0
votes
2answers
15 views

Does the definition of eigenvalues work for non-injective linear maps?

I have this result that says: Let $T:V\to V$ be a linear transformation and let $I$ be the Identity map ($I(v)=v$), then $\alpha$ is an eigenvalue of $T$ iff $T-\alpha I$ is not one to one. My ...
4
votes
1answer
38 views

Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?

Let $A \in {M_n}$ and $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
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1answer
22 views

Is the sum of rank one matrices positive semi definite?

Consider $n$ vectors $x_i$ such that $x_i\in\Re^n,i=1,\dots,n$, the matrix: $$ A=\sum_{i=1}^n x_i x_i^T, $$ is positive semidefinite? Are there any results in literature?
3
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1answer
60 views

How to curve fit an unknown function?

I have data which can be described by $y=f(x,z)$ where $z$ varies from 170 ~ 154. Now values given by $ks$ are known sample values that equals value given in the table header, $uks$ are unknown ...
0
votes
1answer
25 views

Lipschitz constant of L2 difference

What is the Lipschitz constant of $$f(A)=||Ax||_2-||Ay||_2?$$ In particular, is it $||x-y||_2$, i.e. is it true that given $A,B,x,y$, the following inequality holds: $$|f(A)-f(B)|\leq ||x-y||_2 ...
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1answer
26 views

System of equation involving tensor

I have to solve a system of equation involving tensor: \begin{align} \underline{\underline{a_1}}\cdot\underline{x} + \underline{\underline{\underline{\underline{b_1}}}} \therefore ...
1
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1answer
42 views

Decomposition of the image of a projection.

Let $\mathbb K$ be a field and $E$ be a $\mathbb K-$vector space. Let $p$ and $q$ two linear endomorphisms such that $p^2=p$ and $q^2=q$ and $p\circ q=0$ and let $r=p+q-q\circ p$. I want to show that ...
0
votes
1answer
17 views

Proving if a transformation is linear.

I have a terrible understanding of Linear Algebra so I'm trying every resource out there that I can. Hopefully my questions won't come off as idiotic. Anyway, we have a transformation from $R^2$ to ...
7
votes
6answers
856 views

is matrix transpose a linear transformation?

this was the question posed to me. does there exist a matrix $A$ for which $AM$ = $M^T$ for every $M$.the answer to this is obviously no as i can vary the dimension of $M$. but now this lead me to ...
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votes
1answer
21 views

What scaled version of vector to use in QR-factorization when vector subtraction is involved

Im trying to figure out if I understand the conceptual basics. All the time you see that vectors are scaled down/up for readability or for simplifying future calculations with that same vector. As ...
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votes
1answer
22 views

Representation of a matrix as product of unitary matrices and diagonal matrix

Let $C=A+B$ where $A$ is a symmetric positive definite matrix and $B$ is a positive semi-definite skew symmetric matrix. Clearly $C$ is neither symmetric nor skew symmetric.Then is it possible to ...
3
votes
1answer
91 views

Maximize trace of matrix equation given two constraints

Let $\mathbf{Q}$ be a rotation matrix and $\mathbf{A}$ and $\mathbf{B}$ be two real-valued matrices of the same size. I want to maximize the function $$ f(\mathbf{Q})=tr\;\mathbf{QA} \qquad ...
0
votes
0answers
10 views

Diagonalization of an endomorphism

Let $K$ be a field of characteristic $0$ and $K[X,Y]$ be the polynomial ring in two variables. Consider the endomorphism $\Delta \in End_K (K[X,Y]$, $\Delta : f \mapsto X \cdot \delta_X (f) + Y ...
0
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0answers
15 views

Determining Counts of Discrete Objects Using Linear Algebra

I'm teaching myself linear algebra and was able to solve the following question using trial and error, but--how would one setup and solve a question like this using Linear Algebra? I have 32 bills ...
0
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1answer
23 views

how to identify the subspace of vectors? [on hold]

Which of the following subsets of $\Bbb R^3$ are subspaces of $\Bbb R^3$? A. The $3\times 3$ matrices with all zeros in the second row B. The $3\times 3$ matrices whose entries are all integers C. ...
0
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0answers
22 views

How to find eigen vector for an eigen value in generalized eigen value problem

I have a generalised eigen value problem of the form $A$x = λ$B$x. I have computed the eigen value (say λ1) I am interested in using Eigen library(C++). However, because the library does not support ...
0
votes
1answer
19 views

Finding a linear map.

I have some problem with a question related to linear maps. I know the solution but I can not understand the reason behind it. For any polynomial $p∈P^2$ let: ...
1
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1answer
47 views

Proving that the matrix exponential map is surjective onto the general linear group

Let $M_n(\mathbb{F})$ be the set of all $n\times n$ with entries in $\mathbb{F}$ and let $\exp:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be defined by $$ \exp(A)=\sum_{k=0}^{\infty}\frac{A^k}{k!},$$ for ...
4
votes
0answers
58 views

Linear Algebra Proof conformation

I am trying to prove this theorem in my book. Because it is provided without proof, please let me know what you think! $\mathbf{Theorem:}$ Let V be a finite , $n$ -dimensional vector space and let U ...
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1answer
43 views
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0answers
17 views

What is my error in this matrix / least squares derivation?

I'm doing a simple problem in linear algebra. It is clear that I have done something wrong, but I honestly can't see what it is. let, $y = Ax$, $y_{ls} = Ax_{ls}$ where A is skinny, and $x_{ls} = ...
1
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0answers
28 views

Proving the hat functions are linearly independent

$$ H_i(x) = \begin{cases} (x-x_{i-1})/(x_i-x_{i-1}), & x_{i-1}\le x\le x_i, \\ (x_{i+1}-x)/(x_{i+1}-x_i), & x_i\le x\le x_{i+1}, \\ 0, & \text{otherwise}. \end{cases} $$ How can I prove ...
0
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1answer
437 views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
5
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1answer
63 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
0
votes
1answer
10 views

Point within a spherical triangle given areas

Consider a spherical triangle like this: where $A_1, A_2, A_3,$ and $P$ are points on the sphere and $t_1, t_2, t_3$ are the proportion of the area of the large triangle contained within the small ...
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4answers
2k views

Rank of product of a matrix and its transpose [duplicate]

How do we prove that $rank(A) = rank(AA^T) = rank(A^TA)$ ? Is it always true?
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1answer
23 views

Bounded operator on $L^{2}(a,b)$

Let $p\in]1,\infty[$ and consider the mapping $$ T : L^{2}(-2,2) \to L^{2}(-2,2), \quad (Tf)(x):=xf(x)$$ I want to show that $T$ is bounded, $||Tf||_L \leq T ||f||_L $. So, $$ ||Tf||_L \leq ...
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2answers
22 views

Point reflection uniqueness

Suppose we have normed vector space V and mapping R from V to itself satisfying the following properties: 1) R has unique fixed point $~a\in V$ 2) $\forall x\in V ~~~~ |Rx-a|=|x-a|$ 3) $\forall x ...
2
votes
0answers
31 views

How can I solve this system of linear equations?

$$\begin{align} x+y(k^2-6)+z(4k+4)&=5k+3\\ -x+y(2k^2-6)+z(4k+4)&=6k+3 \end{align}$$ I must use matrices to find for which values of $k$ this system has: exactly one solution, ...
0
votes
3answers
52 views

Basic way to show for $n\times n$ matrices $A$ and $B$, that $(AB)^{-1} = (B^{-1})(A^{-1})$

In looking at matrix inverses, I know the following works (I is the identity matrix): If $AB$ are nxn matrices and are invertible, then $(AB)C = I$, and therefore $C = (AB)^{-1}$. I can show that ...