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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3
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1answer
37 views

Determining Degeneracies of Operator (Quantum Mechanics / Linear Algebra)

Hello all! Above is my question. I am fine all the way up to the final part about the degeneracy. I find counting degeneracies quite difficult, and this is no exception! I have really no idea why ...
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4answers
50 views

power series for square root matrix

Suppose I have a matrix of the form $$U\ =\ (I+z\thinspace X)^{\frac{1}{2}}$$ where $I$ is the $n\times n$ identity matrix, $z\in\mathbb{C}$ and $X$ is a $n\times n$ arbitrary complex matrix with ...
21
votes
1answer
227 views

vector spaces whose algebra of endomorphisms is generated by its idempotents

Let $V$ be a $K$-vector space whose algebra of endomorphisms is generated (as a $K$-algebra) by its idempotents. Is $V$ necessarily finite dimensional? EDIT (Jul 26 '14) A closely related question: ...
0
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0answers
29 views

Eigen value 0 and 1 , pick out the correct answer

Let $A$ be a real symmetric $n\times n$ matrix whose only eigenvalues are $0$ and $1$. Let the dimension of the null $A-I$ be $m$. Pick out the correct statement (a) the characteristic polynomial is ...
0
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1answer
14 views

Oblique projection for which the projection vector is at an angle of 45 degrees

dixit: A special case of oblique projection is called cavalier projection. It is given when the projection vector forms an angle of 45° with the z-axis. This means that: $$(x_p^2+y_p^2)/z_p^2=1$$ My ...
0
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1answer
34 views

Does substitute $x$ with matrix $A$ in a polynomial conflict with the Axiom of Substitution?

This seems to be an elementary question, gonna ask it anyway. Suppose that $A$ is a square matrix, and that $p(x)$ is its characteristic polynomial, we know that (1) $p(x) = det(xE - A)$ We also ...
0
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1answer
45 views

Solving set of equations

$$ \begin{array}{rcl} pa_0 + (1-p)a_2&=&a_1\\ pa_1 + (1-p)a_3&=&a_2\\ pa_2 + (1-p)a_4&=&a_3\\ pa_3 + (1-p)a_0&=&a_4\\ pa_4 + (1-p)a_1&=&a_0\\ a_0 + a_1 + a_2 + ...
-1
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1answer
12 views

Usefulness of Laplacians for directed graphs

Are laplacians for directed graphs used in any algorithms ? For example laplacians for the undirected graphs are used in algorithms such as spectral clustering.
2
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0answers
25 views

Finding gradient of an objective as a PDE

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
1
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2answers
41 views

Do projections on $\mathbb{R}^2$ transform straight lines to straight lines?

A linear transformation $P:\mathbb{R}^{2} \longrightarrow \mathbb{R}^{2}$ is called projection if $P \circ P =P$. The question is: If $P$ is a projection then $P$ transforms straight lines ...
0
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2answers
91 views

How to transform between two layout forms of matrix calculus?

I'm trying to derive a very simple matrix derivative : take derivative of Tr(A' X) with respect to X. However, I got two different answers by following different methods. First Method: vec routine: ...
4
votes
1answer
712 views

Derivative of trace of a matrix

$$\dfrac{\partial\operatorname{Trace}\left[\left(AB\right)^{T}Q\left(AB\right)\right]}{\partial A}=\text{ ?}$$ where $Q = Q^T>0$
0
votes
2answers
72 views

Prove that the group $(A,+, ◦) $ is a non-commutative ring

• $A × A → A, (f, g) → f + g$, where $(f + g)(x) = f(x) + g(x)$ for all $x ∈ K$ • $A × A → A, (f, g) → f ◦ g$ where $(f ◦ g)(x) = f(g(x))$ for all $x ∈ K$ Show that $(A,+,◦)$ is a non commutative ...
2
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2answers
34 views

Analytic Geometry: One sheeted hyperboloid

Good afternoon! I have a question about analytic geometry. I don't actually know if the answer is quite simple, and I missed something while revising, or if it is actually more complicated than I ...
0
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0answers
34 views

Differential forms and minor expansion, question about notation.

There are lectures by Theodore Shifrin on differential forms, and sadly one video ends suddendly where he explains some notation. I try to formulate it in my own words: When k=n, we have ...
0
votes
2answers
50 views

When $\sqrt{(x, Ax)}$ is a norm?

In $\mathbb{R}^n$, when can a matrix $A$ be turned into a norm $||x||^2 = (x,Ax)$? I have already realized that when $A$ is symmetric and its eigenvalues are strictly positive, $(x,Ax)$ is a norm. ...
0
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2answers
92 views

A silly problem on equivalent statements of linear dependence

Let $v_1, \cdots, v_n \in V$ where $V$ a vector space. Exhibit that the following are equivalent: (1) The vectors are linearly dependent (2) There exist scalars $a_1, \cdots, a_n$ not all zero such ...
1
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1answer
28 views

What is the SVD of $A^{-1}$?

Let $A\in R^{n\times n}$ with full SVD $U\Sigma V^T$ where $U$ and $V$ are orthogonal $n\times n$ matrices and $\Sigma$ is an $n\times n$ diagonal matrix with entries $\sigma_1 \geq\cdots\geq \sigma_n ...
1
vote
1answer
214 views

Total unimodularity of matrix with consecutive ones property

A matrix has the consecutive ones property (often abbreviated C1P) if its every row (or column, for column-oriented C1P) is of the form $(0,\ldots,0,1,\ldots,1,0,\ldots,0)$. There is a theorem which ...
0
votes
1answer
17 views

Show that $\Vert A\Vert= \sigma_1$.

Let $A\in R^{n\times n}$ with full SVD $U\Sigma V^T$ where $U$ and $V$ are orthogonal $n\times n$ matrices and $\Sigma$ is an $n\times n$ diagonal matrix with entries $\sigma_1 \geq\cdots\geq \sigma_n ...
0
votes
2answers
63 views

Eigenvalues of negative companion matrix

Here's a homework question I've been stuck on for a while. Given $A = \left[ \begin{array}{cccccc} 0 & 0 & 0 & \cdots & 0 & a_0 \\ -1 & 0 & 0 & \cdots & 0 & ...
1
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0answers
27 views

Why is the matrix of a Riemannian metric positive definie?

Maybe I could post this as a linear algebra problem but I'll give some context. I know that if $(U, x_1, \ldots, x_n)$ is a local chart of a smooth manifold $M$ I can write locally a Riemannian ...
1
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1answer
15 views

Can the matrix product $PA$ be skew-symmetric with $P=P^T>0$ and $A$ Hurwitz?

Let a real (square) matrix $\mathbf A$ is Hurwitz (i.e., all the eigenvalues of $\mathbf A$ have negative real parts). And let $\mathbf P$ is a real symmetric positive definite matrix. What will be ...
8
votes
4answers
393 views

What are differences between affine space and vector space?

I know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by ...
1
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0answers
13 views

Multidimensional fitting of two data sets

My problem is the following: A laser gives out a bunch of data points which are reflected off a metal surface and recorded by a camera attached to the side of the laser. The image the camera receives ...
2
votes
1answer
61 views

sign determinant $2\times 2$

I have been reading internet and tried to understand the explanation of the sign of a determinant of a $2\times 2$ matrix. if I have a matrix \begin{array}{cc} a & b \\ c & d\\ ...
4
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0answers
73 views
+50

solving equation also involving unknown matrix in trace

Given two real $m$ x $k$ matrices $A_1$ and $B_1$ and two $k$ x $k$ real matrices $A_2$ and $B_2$ I want to solve the following equation for $Q$. $Q$ is an orthogonal matrix, i.e. $Q^TQ=I$. ...
1
vote
1answer
33 views

Infinite series for recurrence

Question 1 If I define $A(z) = \sum_{n \ge 0} a_n \frac{z^n}{n!} \tag 1$ (where $a_n$ are $3\times 3$ constant matrices indexed with n), then can we re-write $\sum_{n \ge 1} a_{n-1} \frac{z^n}{n!} ...
0
votes
0answers
34 views

ODE with multiple simple conditions $f'(x)=f(x)(Ax+D ) $

I have an ODE to solve . The main issue is,in addition to solving it I have to keep some conditions too in the solution of f(x).. I am bit confused regarding how to deal with it. Equation is given ...
2
votes
1answer
21 views

Linearly independent subset of a spanning set

Given $V_1 + V_2 \in \operatorname{Sp} \{V_1,..,V_n\}$ and $V_1 \notin \operatorname{Sp}\{V_2,...,V_n\}$, prove that $\{V_2,...,V_n\}$ is linearly independent. Well, I know that $ V_1 + V_2 \in ...
1
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0answers
41 views

Eigenvalue formula for 4x4 symmetric matrix

Is there a formula/algorithm that is accurate to used in finite precision arithmetic (aka numerical stable ) for small symmetric matrix of size 4x4. Additionally I'm looking if it require similar ...
7
votes
2answers
263 views

Uncountable Basis?

I was reading up on the difference between countable and uncountable sets, and was wondering if there was a basis of uncountable size. I now know there are, however they all seem to be covering ...
0
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0answers
19 views

Testing for Linear Independence/rank mod m

I am working on cracking a hill cypher using modular linear algebra. Every example I have found online makes a big assumption that is not necessarily the case, and as I see it leaves a lot to be ...
0
votes
1answer
41 views

Interpretation of $(r,s)$ tensor

A tensor of type $(r,s)$ on a vector space $V$ is a $C$-valued function $T$ on $V×V×...×V×W×W×...×W$ (there are $r$ $V$'s and $s$ $W$'s in which $W$ is the dual space of $V$) which is linear in each ...
1
vote
1answer
48 views

How to solve the equation $AX=B$ in Matlab?

I am trying to solve an equation of the form AX=B where A, X and B are following matrices. I have the A and B matrices and I have to find the value of matrix X. How can I find the value of matrix X. I ...
2
votes
2answers
36 views

If the 2-norm of a matrix is small, the trace of the matrix is also small

Is it true that If the 2-norm of a symmetric real matrix is small, then the trace of the matrix is also small? I played around with some matrices in MATLAB and discovered this phenomenon. Does there ...
0
votes
1answer
45 views

Volume of Region in $\mathbb{R}^2$.

Consider $$ S = \left\{(x,y) \in \mathbb{R}^2; -N-\frac{1}2 \le x \le N + \frac{1}2, |\alpha x-y| \le \frac{1}N \right\}$$ where $N \in \mathbb{N}, \alpha \in \mathbb{R}$. I'm having a hard time ...
0
votes
1answer
53 views

Given linear maps $T:V\to W$ and $S:V\to W$ does there exist linear map $F:V\to W$ with ker F=ker $T\cap $ ker S

Given linear maps $T:V\to W$ and $S:V\to W$ does there exist a linear map $F:V\to W$ with ker F=ker $T\cap $ ker S, where $V$ and $W$ are different vector spaces? What if $V=W$? The answer is in ...
2
votes
1answer
91 views

Are all fields vector spaces?

Are $\mathbb{Z_p},\mathbb{Q},\mathbb{R},\mathbb{C}$ above themselves vector space? Is a field above anoother field a vector space? As for 1. we know that $\Bbb R^n$ is a vector space so in ...
0
votes
3answers
34 views

Find a value r so that the vector v is in the span of a set of vectors

Find the value r so that, $$v = \begin{pmatrix} 3 \\r \\-10\\14 \end{pmatrix}$$ is in the set, $$ S= \text{span}\left(\begin{pmatrix} 3\\3\\1\\5 \end{pmatrix}, \begin{pmatrix} 0\\3\\4\\-3 ...
1
vote
2answers
45 views

Find $2\det ( \frac{1}{2} A )$ given that $A$ is $3\times 3$ and $\det(A)= -2$

Here is a question that should be done today: If $A$ is $3\times 3$ and $\det(A)= -2$, find $2\det(\frac{1}{2}A)$. I solved this problem but I am not sure because the way I used is not accurate! ...
1
vote
1answer
43 views

Can I find the minimal polynomial by using the characteristic polynomial?

Let's say I have the characteristic polynomial of an operator: $$p(z)=(z-\lambda_1)^{j_1}(z-\lambda_2)^{j_2}\dots(z-\lambda_n)^{j_n}$$ Wouldn't then the minimal polynomial be exactly: ...
0
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1answer
18 views

Prove that the direct sum of a symmetric and skew symmetric matrix belongs to $M_n(K)$ using $A_{ij}$ and $A_{ji}$ notation.

Basically Let $M_n(K)$ be an $n\times n$ matrix of a $K$ vector space. $U =\{A\in M_n(K)\;|\;A_{ij}=A_{ji}\}$ $W =\{A\in M_n(K)\;|\;A_{ij}=−A_{ji}\}$ So I don't understand my mark scheme. It says ...
0
votes
1answer
341 views

Find an ordered basis $B$ for Mnxn(R) such that [T]$B$ is a diagonal matrix for n > 2?

I have a homework problem that I'm stuck on. It is problem 5.1.17 in the Friedberg, Insel, and Spence Linear Algebra book for reference. "Let T be the linear operator on Mnxn(R) defined by $T(A) = ...
3
votes
1answer
50 views

Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
0
votes
1answer
42 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
0
votes
1answer
38 views

First-order linear differential equation for matrix valued functions of size $3\times 3$

I have two matries given by (M' means derivative w.r.t x) $ M=\left( \begin{array}{ccc} f_1(x) & f_2(x) & f_3(x) \\ f_4(x) & f_5(x)& f_6(x) \\ f_7(x) & f_8(x) & ...
0
votes
1answer
34 views

Prove that $\exists$ U: $P$ is self adjoint if and only if $P=P_U$

Suppose $P \in L(V)$ is such that $P^2 = P$. Prove that there is a subspace U of V such that $P= P_U$ if and only if P is self adjoint. First suppose that $P = P_U$ Show this implies that P is self ...
1
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0answers
25 views

How to see that $\text{dim}(L)=k-1$?

Consider $L:=\left\{x\in\mathbb{R}^k: cx=\delta\right\}$ with $c=(c_1,\ldots,c_k)$ and $\delta\in\mathbb{R}$. Show that $\text{dim}(L)=k-1$. Do not know how to show that. Anyhow my first ...
0
votes
1answer
57 views

$A \cdot B = A \cdot C$ does not imply that $B = C$

I am trying to prove the following: If $A,B,C$ are non-zero vectors such that $A \cdot B = A \cdot C$, then it's not necessarily true that $B = C$. My proposed proof. Suppose $A \cdot B = ...