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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34 views

is there a way to multiply in the following tensor?

I have an $R^{n \times n \times n \times n}$ tensor that maps a matrix to another matrix, call it $K$. I also have the matrix $C = A \times B$ where $C,A,B \in \mathbb{R}^{n \times n}$ and $\times$ is ...
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4answers
66 views

Simplifing logarithmic equation

I have the result of a differential equation to be: $$\ln(x+3)=3\ln(t+2)+C$$ I want this to be as simplified as possible. Can it be proceeded like: $$e^{(x+3)}=3e^{(t+2)+C}$$ I am not sure about ...
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2answers
49 views

Where's the error in my calculation of a line through a point and being the tangent to a circle?

$$C:x^2+y^2=r^2$$ $$A(0,A_y)$$ I'd like to find the line L through A and being a tangent on C. Define point P on C. $$P(P_x,P_y)$$ $$P_x^2+P_y^2=r^2$$ Get the slope of L, by calculating the ...
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3answers
158 views

Prove the inequality

I need to prove that $$\frac{k(k+1)}{2}\left(\frac{a_1^2}{k} + \frac{a_2^2}{k-1} + \ldots + \frac{a_k^2}{1}\right) \geq (a_1 + a_2 + \ldots + a_k)^2\;,$$ where $a_1, a_2, \dots, a_k$ is some set of ...
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3answers
81 views

Show that the dimension of a particular linear space is $2$

Question: A Linear transformation $T: \mathbb R^4 \to \mathbb R^4$ is represented by the matrix $$\mathbf A=\begin{pmatrix} \\1&-1&2&3 \\ 2 & -3 & 4 & 5\\ 5 & -6 & ...
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1answer
206 views

Tensor Algebra of Tensor Algebra

Suppose $V$ is a vector space and $T(V)$ the tensor algebra of $V$. What happens if we take $T(T(V))$ that is the tensor algebra of the (vector space) $T(V)$? I 'guess' I heard that $T(T(V)) \simeq ...
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1answer
116 views

V is a vector space such that $V = A\oplus A^\perp$ also $V = A \oplus C$ then can we say that $A^\perp = C$?

I have a vector space $V$ such that $V = A\oplus A^\perp$ i.e. $V$ is a direct sum of its subspace $A$ and orthogonal complement of $A$. Suppose we also have $V = A \oplus C$ Then can we say that ...
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1answer
250 views

How to show that $N(A) = R{(A^*)}^\perp$ and $N(A^*)=R({A})^\perp$?

How to show that for a given square matrices $N(A) = R{(A^*)}^\perp$ and $N(A^*)=R{(A)}^\perp$ where $N(A) $ and $R(A) $ are the null and range spaces of matrix $A$, respectively? I am not able to ...
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0answers
69 views

is the following correct for tensor products?

Let's say that I have a three dimensional tensor $A = ((B \times_1 C_1) \times_2 C_2) \times_3 C_3$. where $B$ is $n \times n \times n$ tensor, $C_i$ are $n \times n$ matrices and $A$, as a result, is ...
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1answer
922 views

Prove that if a matrix A is symmetric, then it is diagonalisable

I need to prove that if a matrix $A_{2 \times 2}$ is symmetric, i.e., $A^t = A$, then it is diagonalisable. I know that a matrix $M_{n \times n}$ is diagonalisable, if and only if there is a basis of ...
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2answers
60 views

is there a tensor that does the following?

I want a tensor (in the multi-linear algebra sense) which takes as an input a matrix $A$ of size $n \times n$ and returns as output an $n \times n$ matrix which is diagonal (zero off-diagonal), and on ...
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2answers
908 views

How can I normalize a percent to a value while still deriving results from the percent?

My math skills are rusty(at best) and I was wondering if I could pick people's here brains on trying to figure out how to approach what I'm doing. My problem is a bit domain specific so I found a ...
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6answers
1k views

A finite-dimensional vector space cannot be covered by finitely many proper subspaces?

Let $V$ be a finite-dimensional vector space, $V_i$ is a proper subspace of $V$ for every $1\leq i\leq m$ for some integer $m$. In my linear algebra text, I've seen a result that $V$ can never be ...
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1answer
360 views

Given a matrix, find a linear transformation that uses it

The matrix is: $$\begin{pmatrix} 3+l & 8 & 3 & 3+l \\ 8 & 9 & 3 & 7 \\ 3 & 3 & 7 & 8 \\ 3+l & 7 & 8 & 13 \end{pmatrix}$$ I'm given the above ...
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0answers
93 views

distance of an affine subspace to a polytope

I wonder how to prove the following statement. Let $V$ be a $d$-dimensional normed space with $d \geq 3$, let $P \subset V$ be a $(d-2)$-dimensional polytope. Then there is an $\epsilon > 0$ such ...
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1answer
103 views

About $P_{{L},{M}}$, projection transformation onto subspace $L$ along subspace $M$ .

I need help to study following theorem: For every idempotent matrix $E\in\mathbb{C}^{n\times n}$, $R(E)$ and $N(E)$ are complementary subspaces with $E = P_{{R(E)},{N(E)}}$. Conversely, if $L$ and ...
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0answers
70 views

A function $\mathbb{R}^n\to\mathbb{R}^n$ that preserves distances must be a linear map followed by a translation [duplicate]

Possible Duplicate: Are isometric normed linear spaces isomorphic? $ f: \mathbb{R}^n \to \mathbb{R}^m $ preserving distances Consider the set of all functions $\varphi : \mathbb{R}^n ...
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2answers
455 views

$R(AB)=R(A)$ iff rank$(AB)$=rank$(A)$, $N(AB)=N(B)$ iff rank$(AB)$=rank$(B)$

$A$ and $B$ are two square matrices then show that $R(AB) = R(A)$ iff $\mathrm{rank} (AB) = \mathrm{rank} (A)$, and $N(AB) = N(B)$ iff $\mathrm{rank} (AB) = \mathrm{rank} (B)$. Here is my ...
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3answers
59 views

Prove two pairs of subspaces are in the same orbit using dimension

Let $V$ be a finite-dimensional vector space over a field $K$. Consider the group $GL(V)$ of non-singular linear maps acting on pairs of subspaces $(U,W)$ of fixed dimensions $p$, $q$ respectively by ...
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1answer
406 views

Transformation matrix for triangle projection in $xy$ plane

It's been a long time since I did computer graphics and algebra, and I don't remember how to correctly manipulate linear transformations. My scenario: I have 3 points, $a,b,c \in \mathbb{R}^3$, ...
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2answers
506 views

Symmetric diagonally dominant matrix

Suppose I have a real, symmetric, $n\times n$ matrix $A$ such that the following conditions hold: 1) All diagonal elements $a_{ii}$ are strictly positive. 2) All off-diagonal elements $a_{ij}$ are ...
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1answer
164 views

endomorphism as sum of two endomorphisms (nilpotent and diagonalizable)

$V$ is a field over $\mathbb{C}$. Show that $\phi: V \to V$ can be written as $\phi = \psi + \sigma$ where $\psi$ is diagonalizable and $\sigma$ is nilpotent. I managed to show this first part (you ...
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2answers
79 views

Characteristic polynomial of the unique automorphism of the zero module

Is there any convention which makes sense of the characteristic polynomial of the unique automorphism of the zero module? This might seem like an odd question but it matters to me. The background ...
2
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1answer
665 views

Is it possible to determine if this matrix is ill-conditioned?

I want to better understand ill-conditioning for matrices. Say we're given any matrix $A$, where some elements are $10^6$ in magnitude and some are $10^{-7}$ in magnitude. Does this guarantee that ...
4
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1answer
61 views

About the intertwiners of a real representation and its complex conjugate

i am currently trying to understand a proof in Trautman's "The Spinorial Chessboard", namely theorem 4.2 on page 48. It states the following: If $\rho:\mathcal{A}\to\operatorname{End}_\mathbb{C} S$ ...
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2answers
137 views

Why is a matrix $A\in \operatorname{SL}(2,\mathbb{R})$ with $|\operatorname{tr}(A)|<2$ conjugate to a matrix of the following form?

The trace $\operatorname{tr}(A)$ of a matrix $A$ is the sum of its diagonal entries. Apparently if $A\in \operatorname{SL}(2,\mathbb{R})$ and $|\operatorname{tr}(A)|<2$, then $A$ is conjugate in ...
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627 views

Finding Transition Matrix

Problem: Find the transition matrix P such that $P^{-1}AP=B$ where: $$A=\begin{bmatrix} 3 & -1 & 0 \\ -1 & 0 & -1 \\ 0 & 1 & 1 \end{bmatrix} \quad\text{and}\quad ...
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2answers
754 views

Finding Matrix Representation

Problem: Let T: $\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear map given by $$T\left[ \begin{matrix} x\\y\\ z\end{matrix} \right]= \left[ \begin{matrix} 3x-y\\z-x\\z-y\\\end{matrix} \right]$$ ...
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3answers
183 views

Jordan decomposition of $A^T$ given that of $A$

Suppose I have the Jordan normal form of a matrix $A$. The decomposition involves the Jordan matrix $J$ and a similarity matrix $P$ such that $P^{-1}.J.P = A$. My question: is it possible to find the ...
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1answer
123 views

Finding value of a constant in Differential Equations

I have the following ODE Where given is $x(0)=1$: $$(t+3)dx=4x^2dt$$ After separation of variables I got this: $$\frac{-1}{x} = 4\ln(t+3)+C$$ I think this simplifies more as: ...
3
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1answer
262 views

Complex eigenvalues of real matrices

Given a matrix $$A = \begin{pmatrix} 40 & -29 & -11\\ \ -18 & 30\ & -12 \\\ \ 26 &24 & -50 \end{pmatrix}$$ has a certain complex number $l\neq0$ as an eigenvalue. ...
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2answers
500 views

Sign restriction on the Lagrange multiplier? Why?

Say we are given a linear program where the goal is to minimize $c^Tx$ with the constraints $Ax\ge b$. Why is there a sign restriction on the Lagrange multiplier associated with the active constraints ...
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1answer
127 views

Derivative of a matrix

I want to know if this derivative is correct. I have derived this but not sure if this is correct. I think it is but just to confirm ...
0
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1answer
67 views

I need help to understand meaning of certain terms in a theorem

There are certain terms in the following theorem where I am finding difficulty to figure out. I need help. Theorem. Let $\mathbb{C}_{r}^{m\times n}$ denote the set of all complex $m\times n$ ...
2
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2answers
155 views

Characterization of the interior of a convex set

My question is the following: let $K$ be a convex set in $\mathbb{R}^n$ and $x$ an element of the interior of $K$. Can I affirm that there exist $z_1,...,z_n \in K$ linearly independent and ...
2
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0answers
29 views

order of elements in a partition using Maple

I determined this whole partition but I just want to have the finer the partition for example: I have this ...
3
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1answer
368 views

Rotation matrix from an inertia tensor

I have a set of weighted points in 3D space (in fact, a molecule) and I'm trying to align the principal axes of this set with the $x$, and $y$ and $z$ axes. To do so, I've first translated my points ...
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2answers
1k views

Determinant of symmetric Matrix with non negative integer element

Let \begin{equation*} M=% \begin{bmatrix} 0 & 1 & \cdots & n-1 & n \\ 1 & 0 & \cdots & n-2 & n-1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n-1 ...
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1answer
2k views

Finding the norm of vectors

When finding the norm of the vector: Find $\|2w-2y\|$ such that $w=(1/2,3,1)$ and $y=(0,-1,3/2)$. answer: $$\begin{align*} &2(1/2,3,1)= (1,6,2)\\ &2(0,-1,3/2) =(0,-2,3)\\ ...
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2answers
217 views

Interesting Determinant

Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1<x_2<\cdots<x_n$. Define \begin{equation*} A=% \begin{bmatrix} 0 & x_{2}-x_{1} & \cdots & x_{n-1}-x_{1} & ...
5
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1answer
4k views

Finding Eigenvectors with repeated Eigenvalues

I have a matrix $A = \left(\begin{matrix} -5 & -6 & 3\\3 & 4 & -3\\0 & 0 & -2\end{matrix}\right)$ for which I am trying to find the Eigenvalues and Eigenvectors. In this case, ...
3
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1answer
133 views

Matrices of Trace $0$

The set of all $n$-square matrices with trace $0$ is a subspace of the set of all $n$-square matrices. Is there a standard notation and/or name for this subspace?
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1answer
114 views

Reducing the span of vectors

Out of interest what would be the best way to describe the spanning set of vectors a a and b a=(0,3,-2) b=(1,0,0). Apart form a and b, what other vector belongs to the spanning set? Do i have to ...
3
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3answers
204 views

Eigenvalues of a matrix $A$ and Linear Tranformation

Let $M_{2} (\Bbb R)$ denote the set of $2 \times 2$ matrices. Let $ A \in M_{2} (\Bbb R)$ be of trace $2$ and determinant $-3$. Identifying $M_{2} (\Bbb R)$ as $\Bbb R^4$, consider a linear ...
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2answers
280 views

Eigenvalues of a matrix $A$ such that $ A^2=0$.

Suppose the matrix $A$ is a $2 \times 2$ non-zero matrix with entries in $\Bbb C$. Which of the following statements must be true? $PAP^{-1}$ is a diagonal matrix for some invertible matrix $P$ with ...
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1answer
110 views

Systems of linear equations

I am I right on the following: A 2X2 system of equation with no solution will look like this: x+3y=3 2x+6y =-8 1/2 + 3/6 ≠ 3/8 A 2X2 system of equations with in ...
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2answers
85 views

Finding the dot product.

Finding the dot product of $(-2w) \cdot w$ where $w=(0,-2,-2)$ $$ \text{dot product} = \frac{v . u}{\| v \| . \| u \|} $$ So $$-2 (0,-2,-2)=(0,4,4) \\ (0+4+4) = 8 \\ (0,-2,-2)=-4 \\ 8 \times ...
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2answers
263 views

Completeness of normed spaces

As earlier, I have received an answer from this site that Bolzano Weierstrass' theorem is true for finite dimensional normed spaces, but not for infinite dimensional spaces. This, in particular => all ...
2
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0answers
85 views

Need little hint to prove a theorem .

I have an iterative method \begin{eqnarray} X_{k+1}=(1+\beta)X_k-\beta X_k A X_k~~~~~~~~~~~~~~~~~ k = 0,1,\ldots \end{eqnarray} with initial approximation $X_0 = \beta A^*$ ($\beta$ is scalar ...
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1answer
347 views

power series expansion of the square root of a Hermitian matrix

Is there a power series expansion of the square root of a Hermitian matrix, as a procedure to calculate the square root without taking the inverse or diagonalizing the matrix? I find for scalar number ...