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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4 views

project a signal $S(t) = \sum_0^{\infty}A(k)e^{if(k)t}$ to 3d domain $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $

Definitions: $\vec{v}e^{\vec{w}x} = \vec{v}cos(x) + \vec{w}sin(x)$ $ \psi_0(t) = x\hat{i}t + y\hat{j}t + z\hat{k}t$ $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $ where $w$ and $v$ ...
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2answers
36 views

If some power $A^n$ of a matrix $A$ is symmetric, is $A$ necessarily symmetric?

If $A^{n}$ is a symmetric matrix, should I conclude that A is also symmetric?
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0answers
10 views

Can this be expressed in terms of linear constraints?

I'm attempting to find a matrix $X$ that minimizes some function $f(X)$ subject to the constraint that $$ X=W A Z $$ where $A$ is a given non-negative matrix with rows that sum to 1, and $W$ and $Z$ ...
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0answers
13 views

Conditional expectation of a set of Gaussian variables

I was wondering if there is an efficient way to compute the conditional expectation of every element in a Gaussian random vector ? Specifically: For a pair of Gaussian random variables $[x,y]$, the ...
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1answer
22 views

In what condition we have $(K^{-1})^\ast = (K^\ast)^{-1}$?

Suppose $X$ $Y$ are two finite dimensional Hilbert space. Assume $K$: $X\to Y$ is linear. My question is, in what condition of $K$ that $$(K^{-1})^\ast = (K^\ast)^{-1}?$$
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1answer
16 views

Geometric and Algebraic Multiplicity, zero dimensions

The eigenvalues are $\lambda =0$(because we have multiplication here), $\lambda =1$, and $\lambda =2$ for the given characteristic equation, and as (a) states, that $GM\le AM$. Now, I want to know ...
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1answer
19 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
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1answer
21 views

Different representations of a matrix in reduced row echelon form

EDIT: I decided to ask this question after working on this particular problem. I had the stupidity to think that row-reduced = row reduced echelon form. Brain fart, nothing more to see here... ...
2
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0answers
19 views

Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
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0answers
16 views

Maximum in a linear system of equations

I have a system of equations with a tridiagonal coefficient matrix: $$ \alpha_i f_{i-1} + f_{i} + \beta_i f_{i+1} = \Gamma_i $$ , where $i$ goes from 1 to $n$. For a given $M$, what constraints ...
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2answers
29 views

About the elements of a finite subgroup of $\mathrm{GL}(\mathbb{R}^{n})$

Let $G$ be a finite subgroup of $\mathrm{GL}(\mathbb{R}^{n})$. I would like to prove that for every $g \in G$, $\det(g) \in \lbrace -1,1 \rbrace$. Here are my ideas : since $G$ is a finite subgroup ...
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1answer
16 views

Matrix norm and perturbation problem on finite dimensional $V$

Suppose we have a finite dimensional real vector space $V$ equipped with a norm $\|\cdot\|$ given by $$\|x\|^2=x^tXx$$ where $X$ is a matrix and $x\in V$ is in matrix form. Further let us assume that ...
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2answers
30 views

Final transformation matrix

I have a 3d object, to which I sequentially apply 3 4x4 transformation matrices, $A$, $B$, and $C$. To generalize, each transformation matrix is determined by the multiplication of a rotation matrix ...
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1answer
20 views

Matrix inequality $A^2 \succeq A$

If $A$ symmetric positive semidefinite matrix is the following inequality true. If $A \succeq I$ then \begin{align} A^2 & \succeq A \end{align} This is an equivalent of $a^2 \ge a$ is $a \ge ...
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31 views

Prove $A,B$ share an eigenvalue [duplicate]

Let $A, B, C \in M_n(\mathbb{C})$ (not zero matrices) and let $g(x)\in\mathbb{C}[X]$. Let's assume $AC=CB$. Prove that $A,B$ share an eigenvalue. Things I've already proved (followed by the ...
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2answers
33 views

Finding matrix representation of an Ellipsoid

I have a $2$-dimensional ellipsoid centered at $(1,2)$. The axes are parallel to $y=x$ and $y=-x$, and it passes through points $(-1,0)$, $(3,4)$,$(0,3)$,$(2,1)$. I would like to find the symmetric ...
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1answer
16 views

Linear, Squared and Logarithmic scales with given input domain and output range

The input domain is $[12,24]$ and the output range is $[0,720]$. Now I know that with using linear scaling the value $16$ of the input range is mapped to $240$; with using sqrt scaling the same value ...
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0answers
25 views

Rotating one coordinate system about another

I have two coordinate systems: A and B. I also have a point p, whose position relative to ...
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4answers
70 views

Prove that $g(A)$ is an invertible matrix

Let $A\in M_n(\mathbb{C})$ and let $\lambda\in\mathbb{C}$. Prove that if $\lambda$ is not an eigenvalue of $A$ then $A-\lambda I$ is invertible. Moreover, for $g(x)\in \mathbb{C}[x]$, prove that if ...
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1answer
28 views

AX=B in Matlab solution

I am new to MATLAB and wanted to solve a linear equation and come across a problem The matrix A is [ 3 6; 6 14] and B Is [ 5;11] According to ...
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3answers
52 views

How can I find the co-ordinate of where a line intersects a circle?

I was looking to know if there was an equation that would allow me to calculate the co-ordinates of a point on the circumference of a circle where a line intersects it and the center. My diagram ...
12
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4answers
980 views

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
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1answer
36 views

Show that every vector in the null space of $m \times n$ matrix $A$ is orthogonal to every vector in the row space of $A$. [on hold]

How do I show? do I show this by using inner products? What should I define my inner product?
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0answers
26 views

Matrix polynomial [on hold]

Suppose: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a ...
0
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1answer
20 views

Is it possible to determine if a matrix is not diagonalizable via row operations?

Suppose a matrix can be row reduced to the identity matrix, is this enough to say that it is not diagonalizable? If so, what theorem(s) or logic figures this out?
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4answers
42 views

Equivalent definitions of an orthogonal matrix.

I wish to show that the following definitions of an $n \times n$ real matrix $Q$ are equivalent: $QQ^T=I$, $Qx\cdot Qx=x\cdot x$ for all $x\in \mathbb{R}^n$. I found it easy to show that ...
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0answers
13 views

E as an expectation of a quadratic form [on hold]

if E(expectation of quadratic form) is an operator, show that E(AB+C) = AEB + EC. where b and c are variables.
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18 views

Show that every Jordan matrix has a cyclic vector

Is my following reasoning correct? Since an $n\times n$ Jordan matrix has rank $n-1$ (because we can only make the last row the zero row), its geometric multiplicity is 1, which means the matrix has ...
2
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1answer
30 views

Affinity of lorentz transformations

Lorentz transformations are often defined to be linear. But suppose instead we only consider transformations that preserve the spacetime interval. Is it possible to prove that those transformations ...
4
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1answer
32 views

Determinant inequality and positive definite matrix

Let $B$ and $C$ be $n\times n$ hermitian matrices, with $B$ positive definite and $C$ positive semi-definite. (1) Show that $B+C$ is positive definite (2) Show that $\det(B)\leq \det(B+C)$. What ...
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12 views

Let T be the bounded operator and T* be the adjoint operator of T.Show the following. [on hold]

Let T be the bounded operator and T* be the adjoint operator of T. Show the following. 1.||TT||=||T|| 2.||TT||=||TT*||=||T||^2 3.(T+S)=T+S* 4.(αT)=α ̄T (α∈C) 5.(TS)=S T* 6.(T* )* =T
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0answers
17 views

System of Equations for 3-digit number [on hold]

This is a rare word problem where I've had trouble: Find system of equations and use elimination. The sum of the digits of a three-digit number is 9, and the tens digit of the number is twice the ...
4
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2answers
48 views

Which $n$-forms are pullbacks of top forms on $\Bbb R^n$

Let $V$ be a finite-dimensional vector space. I write $F_n(V)$ for the $n$th exterior power of the dual vector space. Which elements of $F_n(V)$ can be pulled back from a top form along a linear ...
2
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1answer
25 views

Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
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1answer
52 views

Verify the following assertion: [on hold]

Suppose that $U=\{(x,x,y,y)\in F^4:x,y\in F\}$ and $W=\{(x,x,x,y)\in F^4:x,y\in F\}$.Then $$U+W=\{(x,x,y,z)\in F^4:x,y,z\in F\}.$$ Not sure how to add these subsets. Please provide explanation.
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1answer
66 views

How can I find $\det(A)/\det(B)$, when individual determinants blow up

I am interested in the quantity: $\frac{\det(A)}{\det(B)}$ of positive definite matrices $A$ and $B$. The problem I am running into now is that for large $A$,$B$, (around $200 \times 200$), the ...
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0answers
20 views

How can I tell if two lines will cross using vectors [on hold]

things I need 1. a visual recpinatation 2. A explanation on how to solve it and 3. a problem to do
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2answers
47 views

Showing that the set of $2 \times 2$ real orthogonal matrices has a particular parameterization

Theorem Every orthogonal matrix in $\mathbb{R}^{2, 2}$ is in the form \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} or \begin{bmatrix} \cos\theta ...
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0answers
30 views

Maximization of sum of convex functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
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0answers
25 views

Can I do Gaussian Elimination on this? (mod 2)

I have this matrix in GF(2): [0, 0, 1, 0] [1, 1, 0, 0] [0, 0, 0, 1] It's not a square matrix but I tried to do Gaussian elimination on it anyway after adding a ...
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Don't understand about how to solve the first equation with Gauss Elimination

Please take a look at the picture. My quesion is: How to solve the first equation with Gauss Elimination that is displayed in the picture. I don't understand HOW.
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27 views

Linear independent vectors

I am stuck in an argument. Is the following true: Let $\{v_1,\ldots,v_k\}$ and $\{u_1,\ldots,u_k\}$ be sets of linear independent vectors. Set $u:=\sum_{i=1}^k \alpha_i u_i$, for some $\alpha_i$. Now ...
2
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1answer
56 views

A question about matrix algebras

Let $A,B \in M_n$, $n \geq 2$. If $A$ and $B$ do not share a common eigenvector, why is $\mathcal{A}(A,B) = M_n$? Notation and definitions: $M_n$: the set of $n \times n$ matrices over ...
2
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1answer
16 views

Determining dimension of a sum of subspaces in terms of a parameter

Problem: Consider the linear subspaces \begin{align*} U = \text{span} \left\{ (1,0,1,0), (1,a,0,a)\right\} \quad \text{and} \quad W = \text{span}\left\{(-1, a, a^2, 0), (0,1,0,-1)\right\} \end{align*} ...
9
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9answers
176 views

Show that $B$ is invertible if $B=A^2-2A+2I$ and $A^3=2I$

If $A$ is $40\times 40$ matrix such that $A^3=2I$ show that $B$ is invertible where $B=A^2-2A+2I$. I tried to evaluate $B(A-I)$ , $B(A+I)$ , $B(A-2I)$ ... but I couldn't find anything.
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How to compute the Eigenvectors for a Markov matrix?

I have the following matrix for which I want to get the Eigenvectors. I know how to compute the Eigenvalues, but when I compute the vectors in the null space of the matrix, I get the wrong answer. ...
3
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1answer
27 views

A linear functional on the space of transformations is basis independent

I've been working on this problem for a bit and am not sure how to proceed: let $V$ be an $n$ dimensional $\mathbb{ R } $-vector space, and denote by $\mathcal{L}(V)$ the space of linear operators $V ...
3
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8answers
87 views

For two vectors $a$ and $b$, why does $\cos(θ)$ equal the dot product of $a$ and $b$ divided by the product of the vectors' magnitudes?

While watching a video about dot products (https://www.youtube.com/watch?v=WDdR5s0C4cY), the following formula is presented for finding the angle between two vectors: For vectors $a$, and $b$, ...
2
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0answers
27 views

Eigenvalues of the product of two “incidence” matrices

I am trying to solve the following problem. Let the following incidence matrix of an undirected graph with four nodes $$ B = \begin{bmatrix}1 & 0 & -1 & 0 & 0 \\ -1 & 1 & 0 ...
3
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4answers
60 views

Does an $n\times n$ matrix $A$ only have an inverse if $rank(A)=n$? If so, why?

I'm currently learning about the rank and inverses of matrices, and every time I attempt to find the inverse of a matrix with a rank smaller than it's number of rows, I find I am unable. One example ...