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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Prove that no two complex matrices have unit commutator

There is a problem given in a representation theory textbook: Prove that for any finite-dimensional complex vector space $V$ there are no $X, Y \in \operatorname{End}V$ such that $[X, Y] = ...
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2answers
7k views

Relation between rank and number of non-zero eigenvalues of a matrix

Let $T : V\to V$ be a linear transformation such that dimension of $\operatorname{Range}(T)= k \leq n$, where the dimension of $V$ is $n$. Show that $T$ can have at most $(k+1)$ distinct ...
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143 views

Clarification for exercise 1-10 in Spivak's Calculus on Manifolds.

I'm asked to prove that if $T:\mathbb{R}^m \rightarrow \mathbb{R}^n$ is a linear transformation, there exists a number $M$ such that $\|T(h)\|\le M\|h\|$ for $h\in \mathbb{R}^m$. I'm not sure ...
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1answer
47 views

Given an integral symplectic matrix and a primitive vector, is their product also primitive?

Given a matrix $A \in Sp(k,\mathbb{Z})$, and a column k-vector $g$ that is primitive ( $g \neq kr$ for any integer k and any column k-vector $r$), why does it follow that $Ag$ is also primitive? Can ...
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1answer
36 views

Basis for $p-1$-dimensional subspace such that projection of a $p$-dimensional triangular matrix is triangular

Given a lower-triangular matrix $L \in \mathbf{R}^{p \times p}$ and vector $v \in \mathbf{R}^p$, how can I construct a basis $B$ for the subspace $S = \mathbf{R}^p / \operatorname{span}(\{v\})$ such ...
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3answers
233 views

Eigenvector basis

I have this matrix: $$ \begin{pmatrix} 3 & \sqrt{2} \\ \sqrt{2} & 2 \\ \end{pmatrix} $$ I found the eigenvalues ( that are $1$ and $4$) but I find it really difficult to find the bases... ...
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1answer
79 views

Function in inner product

I really don't even understand this question ( I guess it just a simple one but I don't understand this function given) Given $V$, an inner product space and function $F\colon V\to V$ such that for ...
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2answers
102 views

Consider a linear map and a base vector a, is the matrix of the linear map diagonal?

Is there a linear map $f: \mathbb{R^{4}} \rightarrow \mathbb{R^{4}} $ and a base vector a, with (f: a ,a ) matrix to be diagonal? I've found this theorem: Let L be the linear transformation ...
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1answer
170 views

Need help to understand a theorem

I have been reading a theorem related with the existence of the outer generalized inverse of a matrix where i have certain difficulties to understand the theorem. Theorem is as follows. Let ...
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40 views

Issues with the value of the last element in Cholesky decomposition

I am trying to calculate the Cholesky decomposition of a precision matrix. I was expecting a Lower triangular matrix $L$ where $L_{ii}>0$ for all $i$. However, the last element in the diagonal is ...
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2answers
130 views

Issue with calculating the cholesky decomposition

I am trying to calculate the cholesky decomposition of the matrix Q= ...
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1answer
599 views

What's the difference between Jordan and Schur decomposition?

They both seems to decompose a square matrix into a upper triangular matrix, but what's the fundamental difference between these two decompositions? Thanks!
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2answers
299 views

Given a matrix A, how can we find C if A = AC - CA?

Give this matrix A: \begin{pmatrix}-25&2&3&-29\\2&7&7&11\\3&7&7&2\\-29&11&2&11\end{pmatrix} How can we calculate C matrix when A = AC - CA without ...
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1answer
855 views

Derivative of Determinant Map

For $ V= ( V_1, V_2) $ and $ W= ( W_1, W_2) $, given a determinant map $ det : R^2\times R^2\rightarrow R$ defined as $ det (V,W)= V_1W_2-V_2W_1$. Then have to find the derivative of the determinant ...
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187 views

Demonstration: If all vectors of $V$ are eigenvectors of $T$, then there is one $\lambda$ such that $T(v) = \lambda v$ for all $v \in V$.

Let $T: V \rightarrow V$ be a linear operator. I need to demonstrate that if all nonzero vectors of $V$ are eigenvectors of $T$, then there is one specific $\lambda \in K$ such that $T(v) = \lambda ...
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1answer
423 views

Difference between Dimension of a Linear transformation (space) and the Dimension of its Column Space?

As my title suggests, what exactly is the difference? The Column Space of a transformation $T: \mathbb R^n \to \mathbb R^m$ is simply the subspace which "contains" all the possible Images, right? If ...
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1answer
49 views

Solve an equation.

Let $\mu_1, \ldots, \mu_n$ be unknowns. Let $C=(c_{ij})_{n \times n}$ be an invertible matrix. Suppose that $\sum_{\beta=1}^{n} \mu_{\beta} c_{\alpha, \beta}=\pi i$. I think that we can solve this ...
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4answers
457 views

Can every nonsingular $n\times n$ matrix with real entries be made singular by changing exactly one entry?

I was just thinking about this problem: Can every nonsingular $n\times n$ matrix with real entries be made singular by changing exactly one entry? Thanks for helping me.
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1answer
133 views

Generating vector spaces according to some sets that shall serve as a basis

Give a family of elements $\left(v_{1},\ldots,v_{n}\right)$ (where $v_{1},\ldots,v_{n}$ are just some sets), how many vector spaces are there, such that this family is a basis for that vector space ? ...
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2answers
189 views

Given a matrix $A$, is there a matrix $C$ with $AC = CA + A$?

Given this matrix A \begin{pmatrix}7+a&2&3&3+a\\2&7&7&11\\3&7&7&2\\3+a&11&2&11\end{pmatrix} where $a \in \mathbb{R}$ Is there a matrix $C \in ...
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3answers
105 views

Finding the eigenvalues

Given a $3 \times 3$ matrix with real entries such that $\det (A) = 6$ and $\mathrm{trace}( A )= 0$. If $\det (A+I)=0$ where $I$ is a $3 \times 3$ identity matrix , then eigenvalues of $A$ are $-1$, ...
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0answers
155 views

Matrices made negative semidefinite, but not simultaneously

Consider matrices $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, such that $A$ has at least $1$ strictly positive eigenvalue. Let $X_1, X_2 \in \mathbb{R}^{n \times n}$ such that ...
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2answers
70 views

The typical form of an isomorphism between a $K$-vector space and its coordinate space $K^n$

Let $V$ be an arbitrary vector space of finite dimension $n$ over the field $K$. It is known that in that case $$ V\simeq K^{n}. $$ The canonical isomorphism which achieves this is ...
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2answers
38 views

Maximizing the time we reach to a threshold in a series of numbers

I have a problem and I really don't know what kind of mathematical method should I apply to solve or model my problem. I would be thankful If anyone can give me some answer or help. Suppose we have ...
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0answers
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
204 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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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
920 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 ...
2
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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
904 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 ...
8
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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
454 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 ...
2
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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
404 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
505 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
162 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
78 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
660 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 $\rho:\mathcal{A}\to\operatorname{End}_\mathbb{C} S$ ...
12
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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 ...