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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Can a Matrix with positive entries have a negative eigenvalue?

It seems intuitive to me that the answer is no, but I can't prove it.
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1answer
72 views

Algorithms For Large-Scale $\ell_{\infty}$ Minimization

The general problem I want to solve is well studied: $$ \min_x \Vert Ax\Vert_\infty \;\;\; \mathrm{s.t.} \;\;\; Bx=c, $$ which is equivalent to the following linear program: $$ \min_{t,x} \, t \;\;...
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1answer
21 views

Applied Mathematics Question - 30 Lightbulbs; Desynchronization and resynchronization

I have 30 lightbulbs. I want each light bulb to switch on and off with almost imperceptibly different tempos. For example: Lightbulb A is on for 1 second, and then off for 1 second, and then back on ...
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1answer
35 views

Order on eigenvalues on diagonal matrix

If the eigenvalues are say $-1$, $-1$ and $2$ for a $3$ x $3$ matrix, then when comes to the diagonal matrix, is it (from top left, to bottom right) $-1$ $-1$ $2$ or $2$ $-1$ $-1$ or $-1$ $2$ $-1$? ...
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1answer
29 views

Nonempty subsets of an inner product space

I have the following problem about inner product spaces: Let $A$ and $B \supset A$ be nonempty subsets of an inner product space $X$. Show that: a) $A \subset A ^{\perp\perp} $ b) $B^\perp \subset ...
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2answers
71 views

When do two linear hermitian operators have a common eigen vector?

Let $H$ be an finite dimension hilbert space. Let $L_1$ and $L_2$ be two hermitian linear operators acting on this space. I know if these two operators commute they can be diagonalized in a common ...
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2answers
103 views

Find the eigenvalues for a matrix which is a product of matrices

Suppose I have a matrix $A \in \mathbb{R}^{2, 2}$ which is the product of $3$ other matrices, lets call them $A_1 = \left(\begin{matrix} cosx & -sinx \\ sinx & cos x\end{matrix}\right)$, $A_2= ...
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1answer
62 views

Define the trace of T as the trace of the matrix T and prove it.

Recall that the trace of a square matrix A is the sum of the diagonal entries of A. 1) Let T : V → V be a linear operator on V. Define the trace of T as the trace of the matrix of T, represented in ...
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1answer
200 views

Linear algebra does the given line intersect plane?

Determine whether the line $x = (-1, 0, 1) + t(1, 2, 4)$ intersects the plane $2x-y+z=5$. Find the point of intersection if they intersect. I know the equation follows the form $x = p + td$, so I ...
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1answer
55 views

Mass of a wire: intersection of surfaces

So I got this mass problem to solve: Find the mass of the wire formed by the intersection of two surfaces whose density is $\phi=x²$ $\underset{C}\int \phi ds $ along the curve: $$ C:\left\{ \...
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1answer
32 views

existence of $\lambda$ in $V^*$ in an $n$ dimensional vector space

Let $V$ be an $n$-dimensional vector space, and $U \subset V$ a subspace of dimension $n−1$. 1) Show that there exists $λ∈V^∗$ with $\ker(λ)=U$. 2) Show that if $μ ∈ V^∗$ is another linear ...
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3answers
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Derivative of $f(A) = \|A x\|^2$ with respect to the Matrix

Suppose I have $A \in \mathbb{R}^{n^2}$ and $x \in \mathbb{R}^n$ where $A$ is interpreted as a matrix. We can define $f(A) = ||A x||^2$ for some constant $x$. What is the derivative of $f$, written ...
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1answer
18 views

Find set of solutions $S_z:=\{y \in\mathbb{R}^{N}: y'z=\iota_N ' z\}$, $ z\in\mathbb{R}^N$.

How to characterize $S_z:=\{y \in\mathbb{R}^{N}: y'z=\iota_N ' z\}$, $ z\in\mathbb{R}^N$? Is there also a general way for more complex equations $y'\beta(z)=\iota'z$ where $\beta(z)\in\mathbb{R}^N$ ...
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0answers
48 views

Geometric interpretation of Linear Independence and absolute value

I am wondering something in regard to linear independence / dependence of functions. I understand when working with vectors in $\mathbb{R^n}$ for example, we can make the geometric link that vectors ...
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0answers
24 views

Linear Algebra - Linear Combinations

We are asked the following: The span of $$v_1,v_2,…,v_r$$ can be the whole $$V,+,⋅$$ or a subset of $$V,+,⋅.$$ Give an example of each situation. The span is denoted $$span v_1, v_2, ..., v_r $$ ...
2
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1answer
64 views

Can the terms in a 3x3 determinant be any six nonzero numbers?

Given six nonzero real numbers $x_1,\ldots x_6$, can you construct a 3x3 matrix such that the six diagonal products that appear in the determinant are $x_1,\ldots,x_6$, respectively? In other words, ...
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2answers
635 views

Find the point on the line $y=2x+1$ that is closest to the point $(5,2)$.

I already attempted this problem this problem with the formula for Vector projection $(x^\text{T}y)/(y^\text{T}y) y$ and go the solution $(1.1,3.3)^\text{T}$ but the book states the solution is $(1.4,...
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5answers
143 views

When will $\operatorname{det}\left(A\cdot A^{\top}\right)=0$?

I am writing a small computer program to solve certain linear algebra equations as part of a larger program. For two of my functions I need to evaluate $\left(A\cdot A^{\top}\right)^{-1}$. This got ...
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1answer
65 views

W is T-invariant. Define $\bar T: V/W \to V/W, \bar T(v+W)=T(v)+W$.Prove if $T_W$ and $\bar T$ are diagonalizable without common eigenvalue, then is T

$T$ is a linear operator on a finite dimensional vector space $V$, and $W$ be a $T$-invariant subspace of $V$. Define $\bar T: V/W\to V/W$ by $\bar T(v+W)=T(v)+W$. It can be proved that $\bar T$ is ...
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2answers
83 views

Compute power of a matrix $A$ as $n\rightarrow \infty$

We are given $A^p=A ...A$(p times) And we are given matrix A: $A=\begin{vmatrix}0.6&-0.4&0\\-0.4&0.6&0\\0&0&0.5\end{vmatrix}$ I need to compute $A^p$ as p approach Infinity. ...
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2answers
2k views

How to show whether 3 planes have a common line of intersection

To show whether or not the 3 planes $$x+y-2z=5\tag 1$$ $$x-y+3z=6 \tag2$$ $$x+5y-12z=12 \tag 3$$ all have a common line of intersection. Can I do $(3)-(2)$ to get the line $6y-15z=6$ and $(1)-(2)$ ...
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0answers
81 views

How to calculate effect of different variables/parameters on a quantity?

I am developing a game for iOS. In the game I have around eight different parameters that directly affect the score of the player. We can say that these eight variables decide the difficulty of the ...
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1answer
71 views

Conditional Inverse: Find set of solutions for $AXA=A, A:=\left(1_N-\alpha\iota_N '\right)$.

Given $\alpha \in\mathbb{R}^{N}$ with $\alpha'\iota_N=1$, how can I characterize the set of conditional inverses (or c-inverse) of $\left(1_N-\alpha\iota_N '\right)$ defined as: $\{K\in\mathbb{R}^{N \...
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1answer
46 views

suppose $tr(A^k) = tr(B^k)$ for all $k$=$1,2,…$. why $A$ and $B$ are same characteristic polynomial? . [duplicate]

Let $A,B \in {M_n}$ and suppose $tr(A^k) = tr(B^k)$ for all $k$=$1,2,...$ . Why do $A$ and $B$ possess the same characteristic polynomial?
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2answers
41 views

Find values of $t$ so that this matrix is positive definite

I will start from this point: $\det{\left(B-\lambda I\right)}=0\Longleftrightarrow\begin{vmatrix}t-\lambda&3&1\\3&t-\lambda&0\\1&0&t-\lambda\end{vmatrix}=0$ Now we will ...
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1answer
45 views

Zero characteristic polynomial?

Is it possible that characteristic polynomial of an $n \times n$ matrix be the zero polynomial? If this happens, this means that any scalar would serve as an eigenvalue?
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1answer
47 views

Problem on eigenvalues

Let A be real square matrix of order $n \geq 2$. Then show that: A. if $A^3 - I$ is singular, then $1$ is eigenvalue of $A$ B. if $A$ is singular, then $I+2A+A^2$ has eigenvalue $1$ My ...
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0answers
75 views

Abstract interpretation of isomorphism between tensor product with dual and hom

I'm interested in the following statement, coming from Remark 6.4.21 of Qing Liu's Algebraic Geometry and Arithmetic Curves: Let $\mathcal{F}, \mathcal{G}$ be quasi-coherent sheaves on a scheme $X$...
4
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1answer
89 views

How is the perturbation in one column of a symmetric matrix reflected in its eigenvalues?

Suppose we have a 0-1 square symmetric matrix. Then it's eigenvalues are real. But I have observed that by multiplying any of its column by a positive constant, even if the matrix is not symmetric ...
13
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4answers
248 views

Matrices such that $M^2+M^T=I_n$ are invertible

Let $M$ be an $n\times n$ real matrix such that $M^2+M^T=I_n$. Prove that $M$ is invertible Here is my progress: Playing with determinant: one has $\det(M^2)=\det(I_n-M^T)$ hence $\det(M)^2=\...
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2answers
23 views

Special method of solution for $A\vec x=\vec b$ where $A$ is a square matrix such that $A^tA$ is diagonal and has full rank?

Is there any special shorter method of solution other than cramer's rule for solving a system of $n$ linear equations in $n$ unknowns $A\vec x=\vec b$ where the square matrix $A$ has the property that ...
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1answer
120 views

Finding the equation of the new plane after the original has been rotated by an angle

Find the equation of the plane obtained after rotating the plane $x+y+z=1$ by $90^{\circ}$ about its line of intersection with the plane $x-2y+3z=0$. Since I had to choose one of the four given ...
3
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0answers
49 views

Gram-Schmidt in characteristic two?

I was helping someone work on a computing problem with bit vectors that reduced to finding a basis knowing a spanning set, and realized quickly that the Gram-Schmidt process does not work as expected ...
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1answer
36 views

Roots of Matrices and Diagonalization

Question: For which of the following matrices $A_i$ is there A complex matrix $B$ such that $B^2 = A_i$; A self-adjoint complex matrix $B$ such that $B^2 = A_i$; A real matrix $B$ such that $B^2 = ...
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1answer
47 views

Why : if the matrix $A$ is not invertible, then $L_A$ is not onto.

I was reading a book and the following statement was made: If $A$ is not invertible, then $L_A$ is not onto. Here, the matrix $A$ is $n \times n$ I'm just curious as to why this is true. Thank you!...
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3answers
59 views

addition of two vectors in $\mathbb R^2$

Confused about this question ..vectors in space $\mathbb{R}^2$ $$M = \{(x,y)∈ \mathbb{R}^2 | \ 0 ≤ x ≤y\}$$ $$N = \{(x,y)∈ \mathbb{R}^2 | \ 0 ≥ x ≥y\}$$ $$P= M∪N$$ ($P$ is closed under scalar ...
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1answer
44 views

Prove that $A$ diagonalizable.

Let $A$ be an $n \times n$ matrix, and let $v_1,...,v_n$ be a basis of $R^n$ so that each $v_i$ is an eigenvector of $A$. Prove that $A$ diagonalizable. Does the diagonalization of $A = QDQ^{-1}$ ...
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1answer
16 views

$\{L(v_1),…,L(v_2)\}$ is a basis for $V_2$

Prove that if a linear map $L: V_1 \to V_2$ is an isomorphism and ${v_1,...,v_n}$ is a basis for $V_1$ then $\{L(v_1),...,L(v_2)\}$ is a basis for $V_2$. Explain why this shows that the dimension of $...
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2answers
45 views

Is $f \mapsto 2f^1$ linear, where $f^1(x) = f(x+1)$?

The book I got this exercise from says it is, but I don't understand why. Doesn't it not satisfy $f(x+y)=f(x)+f(y)$? I would ask my lecturer, although I'm trying to teach myself this topic from a ...
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1answer
28 views

Can we ensure convergence for the jacobi method or do we simply trial and error?

For iterative methods for solving systems of equations, we may not always get convergence and it can depend simply on the way in which we write the equations. I understand there are tests which will ...
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1answer
73 views

Matrix with unit determinant as a product of elementary matrices.

There are three types of elementary matrices: Type 1: matrices obtained by interchanging the ith row of $I$ and jth row of $I$; Type 2: matrices obtained by multiplying the ith row of $I$ by $\...
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1answer
19 views

$y^tv<0,z^tv<0 \text{ unsolveable} \Leftrightarrow \exists \lambda\geq 0: y=-\lambda z$

I am trying to show: Let $y,z\in\mathbb{R}^n$ and $z\neq 0$. Then $y^tv<0,z^tv<0 \text{ unsolveable} \Leftrightarrow \exists \lambda\geq 0: y=-\lambda z$. '$\Leftarrow$' is trivial. '$\...
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1answer
59 views

$n \times n$ matrix Identity Matrix?

Can anyone explain this conceptual problem? If an $n \times n$ matrix is in row reduced echelon form, explain why it is either the identity matrix or else has a row of zeroes? Thanks
2
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1answer
101 views

Largest entry in symmetric positive definite matrix

I know why in a symmetric positive definite matrix every entry on the trace is positive entry $a_{ii}>0$. However I don't how to show that the largest value of the matrix is also on it's trace, ...
0
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1answer
33 views

Which of the following is not necessarily true $(JNU)$

Question : Suppose $T : \mathbb R^n \rightarrow \mathbb R^n$ is a linear transformation and $T$ has non zeroes distinct eigen values. Then which of the following is not necessarily true ? (1) There ...
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0answers
57 views

Finding basis for image of Linear Transformation

I have been looking up how to find the basis of an image online. I found a solution on StackExchange here It seems that to find the basis we reduce the rows to row echelon form. Then we find the ...
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1answer
60 views

Does $A^{-1}$ exist?

Suppose A is similar to the matrix B given below. $$ B= \begin{bmatrix} 7 & 0 & 0 \\ a_{21} & 4 & 0 \\ a_{31} & a_{32} & -0.5 \\ \end{bmatrix} $...
2
votes
1answer
50 views

Does multiplication by the inverse of a Cholesky matrix preserve order?

Let $n \in \{1, 2, \dots\}$ and let $C \in \mathbb{R}_{n, n}$ be a real, symmetric and positive definite $n \times n$ matrix. Define $B \in \mathbb{R}_{n, n}$ to be the real, lower triangular matrix ...
0
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1answer
31 views

Checking vectors for subspaces in $\mathbb{R}^3$ space.

How to check if these sets are subspaces in $\mathbb{R}^3$ ? i know the three condtions but how to check those conditions with some solvings? Thanks in advance...... $$U_1 = \{(x,y,xy)\mid x,y ∈ \...
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2answers
38 views

exponential functions.

I am confused of solving expnential functions they look easy but cant solve it. 1: $$\large e^{8\cdot\ln(b^{1/4})}$$ and this one solving for x: 1: $$\ln(6x-2) = 5$$ FYI : Its not an assignment. i ...