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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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 ...
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4 views

Show that ||A|| = $\sigma_1$

Given that A $\in R^{n*n} $ with full SVD U$\Sigma V^T$ where U and V are orthogonal $n$x$n$ and $\Sigma$ is $n$x$n$ diagonal matrix with entries $\sigma_1 >= ... >= \sigma_n >= 0$ Show that ...
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2answers
61 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 & ...
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1answer
38 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 ...
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19 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 ...
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1answer
13 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 ...
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12 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 ...
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4answers
145 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 ...
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4 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.
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12 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
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1answer
59 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\\ ...
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66 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. $Q^TQ=I$. ...
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1answer
32 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!} ...
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31 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
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1answer
20 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 ...
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0answers
39 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 ...
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2answers
256 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 ...
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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 ...
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1answer
39 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 ...
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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 ...
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2answers
35 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 ...
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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 ...
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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
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1answer
90 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 ...
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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 ...
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2answers
44 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! ...
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1answer
42 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: ...
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1answer
17 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 ...
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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) = ...
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1answer
47 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 ...
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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
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1answer
37 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
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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 ...
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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 ...
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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 = ...
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2answers
75 views

Prove that $T-\sqrt{2}I$ is invertible.

Suppose $T \in L(v)$ is such that $\|Tv\| \leq \|v\|, \forall v \in V$. Prove that $T-\sqrt{2}I$ is invertible. I know that I need to show there exists a $R^{-1}(T-\sqrt{2}I) = I$ where $R = ...
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2answers
50 views

Solving Systems of Equations..!! [on hold]

I actually don't want to solve the systems of equations. I just want to let them equal each other so I can cancel one of them and solve the other one. Example: $E_1\quad x-y=0$ $E_2\quad 2x-2y=0$ ...
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+500

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: ...
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3answers
96 views

Solution of a recurrence with varying coefficient

Please help me to find the solution of the recurrence in terms of n(implies $(f(n))$ and also the summation of the recurrence up to infinity ($sum = \sum_{n=0}^\infty f(n)$) . ...
21
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10answers
12k views

How to show that $\det(AB) =\det(A)\det(B)$

Given two square matrices $A$ and $B$, how do you show that $\det(AB) = \det(A)\det(B)$, where $\det(\cdot)$ is the determinant of the matrix?
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2answers
597 views

Volume of a Pyramid Linear Algebra

Find the volume of a pyramid with triangular base bounded by vectors (1,-1, 2) and (1, 1, 1) and vertex located at (3, 2, 5). I am not sure how I would solve this. I know the volume of a pyramid is: ...
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2answers
40 views

Show that 2 matrices belong to a square matrix by taking the transpose. Vector spaces

Let $M_n(K)$ be an $n\times n$ matrix of a K vector space. \begin{align} U &= \{A ∈ M_n(K) | A_{ij} = A_{ji} \} \\ W &= \{A ∈ M_n(K) | A_{ij} = -A_{ji} \} \end{align} Prove that $U$ and $W$ ...
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1answer
217 views

Subspaces, transformation matrices exercise

I have trouble understanding the following exercise so I would really appreciate any help you could give me: Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Let $H$ be ...
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36 views

Sparsity of Linear Diophantine Equations

If you are looking for integer solutions to the system. $$Ax=b$$ where $A$ is an integer matrix and $b$ is integer vector, then you can construct the solution space integer matrix $B$ and integer ...
2
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1answer
49 views

Is it true that for an inner product space a norm of a vector is defined unambiguously?

Suppose we have some vector space $V$ for which we have defined an inner product $\langle \cdot\rangle$. Thus we have an inner product space. Is it true that $\forall x \in V : \lVert x\rVert := ...
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2answers
52 views

How to find this Linear Transformation

Q. Find the Linear Transformation $T:V_3\rightarrow V_3$ , such that $T(0,1,2)=(3,1,2)$ $T(1,1,1)=(2,2,2)$ I tried considering $(0,1,2),(1,1,1)$ as basis, it doesnt seem to work that way. Just need ...
1
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2answers
45 views

the rank of a linear transformation

Let $V$ be vector space consisting of all continuous real-valued functions defined on the closed interval $[0,1]$ (over the field of real numbers) and $T$ be linear transformation from $V$ to $V$ ...
5
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2answers
1k views

Is there a unique solution for this quadratic matrix equation?

The quadratic matrix equation I've been looking at lately: $$ Q_{r,r}=A_{r,r}X_{r,r}^2+B_{r,r}X_{r,r}+C_{r,r}=0_{r,r} $$ Note that $A, B, C,$ and $X$ are $r \times r$ matrices. $A$ contains known ...
0
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2answers
19 views

Matrix representation of a linear operator

As I'm studying for my final, my book keeps skipping alot of steps and I don't know how tthey get from point a to point b - probably because its elementary at that stage in the book, except not to me ...
0
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0answers
12 views

How to decompose the given matrix by Geometric mean decomposition??

$$H=\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}.$$ How to decompose this matrix using Geometric mean decomposition?? If anybody knows this method kindly update how to ...