Questions tagged [linear-algebra]
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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.
106,711
questions
0
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4 views
Existence of a matrix similar to upper triangular matrix.
Suppose that $X \in M_{n\times n}(\mathbb R)$, then there exists a nonsingular $S \in M_{n \times n}(\mathbb R)$ s.t $$S^{-1}XS= \begin{pmatrix}
P_{1} & & & \\
&P_{2} &\ &...
0
votes
0answers
18 views
Show that orthogonal projection of $(0,1,0)$ over $W$ is the vector $(\frac{1}{\sqrt{2}}, \frac{i}{2}, \frac{1+i}{4})$
Given $W = Sp\{(1,i,0),(1,2,1-i)\}$ a subspace of $\mathbb{C}^3$ with the standard inner product.
Show that the orthogonal projection of $(0,1,0)$ over $W$ is the vector $(\frac{1}{\sqrt{2}}, \frac{i}{...
0
votes
1answer
16 views
A question about linear application
For example, we have two vector space V, W, and X is a K-base of V.
So, therefore if we have $f,g: V \to W$ two K-linear applications. Suppose f(x)=g(x) for all $x \in X$, how to prove $f=g$?
And for ...
1
vote
0answers
10 views
How to identify generator matrices for a linear code?
Lets say we have some linear code of length n over F2 that is given by:
{a, b, c, d, e, f, g, h}
where a, b, c, d, e, f, g, h represent some element/word in this code and each has length n.
If I were ...
2
votes
2answers
54 views
Can there be a third type of unit other than 1 and i?
I know the question sounds vague, so let me try to explain what is on my mind. When we imagine the complex plane, one axis represents the real part and the other axis represents the imaginary part. ...
-2
votes
1answer
13 views
How to find out the magnitude of addition of three vectors?
There are three unit vectors.
The question is to find the modulus of the addition of these three vectors:
A=3i-2j+k
B=2i-4j-3k
C=-i+2j+2k
2
votes
2answers
38 views
Prove that exists an inner product such that $\{1+x,x+x^2,x^2+x^3,x^3+x^4,x^4-1\}$ is orthogonal
I've been asked to prove or disprove that exists an inner product in $\mathbb{R}_5[x]$ such that
$A = \{1+x,x+x^2,x^2+x^3,x^3+x^4,x^4-1\}$
is orthogonal.
How does one approach this?
0
votes
2answers
22 views
Question about a statement involving system of linear equations and rank of matrix.
the statement is:
let $A\in \mathbb{R}^{m\times n}$, if for every $c\in \mathbb{R}^m$ there exists a solution for $Ax=c$, then $\operatorname{rank}(A)=m$.
Now I can understand that this is a true ...
0
votes
0answers
9 views
Derivative of $vec(XX^T)$ (vectorized factor matrix)
I'm looking for the gradient of a factor structured matrix generated by $X X^T$, subject to some constraints, and I'm very thankful if someone could help me further.
Specifically, I have a Lagrangian ...
0
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0answers
6 views
Show that $M$ and $D$ commute, where $J$ is a matrix in Jordan canonical form, $D$ is its diagonal entries and $M = J - D$.
Question:
Be $J$ be a finite dimensional matrix in Jordan Canonical form, let $D$ be the diagonal matrix with diagonal entries equal to $J$, and let $M := J - D$
Show that $MD = DM$
Attempt:
Let $(A)_{...
1
vote
1answer
34 views
Why does this converge to eigenvector?
I'm currently working on a way to analytically find eigenvectors. Teacher says to show that
$ x_{k+1}\gets \dfrac{Ax_k}{||Ax_k||}$
where $x_{k}$ is the vector and $A$ the matrix converges to an ...
0
votes
0answers
13 views
Explaining the proof of the condition number in relation to the norms of the matrices.
We have a system of equations, $A \vec{x} = \vec{b}$, to which we introduce certain disturbances and get $A (\vec{x} + \delta \vec{x}) = (\vec{b} + \delta \vec{b})$. We can then subtract the second ...
0
votes
0answers
8 views
Right form for my Jacobian with only one function
I am faced to what seems to be simple :
I have a function $\alpha_(b_{1,1},b_{2,1}) = \dfrac{b_{1,1}}{b_{2,1}}$.
I mean that I have only one single function depending of 2 variables $b_{1,1}$ and $b_{...
0
votes
1answer
20 views
Prove equality of linear maps
Let $V,W$ be $K$-vector spaces and $S$ a $K$-basis of $V$. Suppose that $V$ is ļ¬nite dimensional.
Let $\varphi_1,\varphi_2: V ā W$ be $K$-linear maps. Suppose $\varphi_1(s) = \varphi_2(s)$ for all $s ...
0
votes
1answer
8 views
Finding an orthonormal basis z of $\Re^n$ with respect to which the matrix $A_z$ is diagonal
Let $\textbf{A}:\Re^n \rightarrow \Re^n$ be given by $\textbf{A} \vec{x}=\vec{x}+l(\vec{a},\vec{x})\vec{a}$ where $||\vec{a}|| = 1, \vec{a} \in \Re^n, l \in \Re.$ And let $\textbf{A}$ be orthogonal ...
1
vote
1answer
30 views
Factorization of a unimodular matrix
Let $S = \big(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \big)$ be a matrix with determinant $ad - bc =1$ (this is the case I'm interested in, but I don't think it is essential).
...
4
votes
1answer
73 views
Determinant of a sum of square matrices
Let $$A=\begin{bmatrix}0&1&0&\cdots 0\\
0&0&1&\cdots 0\\
\vdots\\
0&0&0&\cdots 1 \\
1&1&1&\cdots1
\end{bmatrix}_{n\times n}$$
i.e. it has ones above ...
0
votes
1answer
17 views
$T$ injective equivalent to $T'$ surjective
I'm currently going through Sheldon Axler's "Linear Algebra Done Right" (3rd Ed.) and struggle with the result of 3.110, where he shows that $T\in\mathcal{L}(V, W)$ is injective if and only ...
0
votes
1answer
25 views
Existence of a non-zero T- invariant subspace on $\mathbb{Q}^{4}$
For any linear transformation $T:\mathbb{Q}^{4} \to \mathbb{Q}^{4} $, does there always exist a non-zero T- invariant subspace?
As we know
For any linear transformation $T:\mathbb{R}^{4} \to \mathbb{...
0
votes
0answers
13 views
Checking/testing bilinearity of a function
I have already aware of the bilinear form of a function which is:
<x,y> =x^tAy= ā_(i,j)=ća_(i,j)x_iy_ić
Applying this concept to an example:
Ī±(x,y)=x^T* [2 -1, *y
-1 1]
Assuming x = [x1 y1], ...
0
votes
1answer
41 views
If you have a set of eigenvectors with full multiplicity, is it always possible to orthogonalize them?
a) construct a (diagonalizable) 2Ć2 complex symmetric matrix not admitting an orthogonal basis of eigenvectors
b) construct a 2Ć2 complex symmetric matrix which cannot be diagonalized
The point of ...
0
votes
2answers
31 views
Dimension of Example Quotient Space
Let $V$ be the vector space across the set of all integers, with subspace $W$ as the vector space across all even numbers, both with the field as the integers. The quotient space $V/W=\{v+W|v\in{V}\}=\...
-3
votes
1answer
23 views
Finding the determinant of unknown matrix
How can I solve this question:
given that $A$ and $B$ are both $3\times 3$ matrices and $\det(A) = -44$ and $\det(B) = -2$.
How can I find the determinant $\det(2A^{T}B^{-1})$?
2
votes
0answers
19 views
Using Cholesky Decomposition iteratively to find eigenvalues
I have an algorithm that should theoretically find eigenvalues of any symmetric, positive definite matrix $A$.
Start with a seed $A_0 = A$ and then find its cholesky decomp, so that $A = U^T U$
then ...
0
votes
0answers
11 views
Analytic methods of computing entrywise absolute values of matrices
Are there any analytic ways of computing, or closely approximating the entry wise absolute value of a matrix?
More explicitly, for an arbitrary matrix $B\in \mathbb{R}^{m\times n}$, are there any ...
0
votes
1answer
19 views
Is there any situation where the LDU decomposition is the same as the eigenvalue decomposition?
I was just wondering if there are any situation where the LDU decomposition is the same as eigenvalue decomposition (diagonalization)? The only way this can be possible if L and U are inverse so ...
0
votes
0answers
20 views
Set of all p-times differentiable functions is vectorspace
I'm stuck at the very basic and trivial task which is to prove that the set $M:=C^p(a,b)$ of all p-times differentiable functions $f: (a,b) \rightarrow \mathbb{R}$ form a vector space, whereas the set ...
0
votes
0answers
22 views
Geometry of vanishing lines
In page 220 of Multiple View Geometry in Computer Vision is the following figure:
So in this figure, $\mathbf{l}$ is the vanishing line of the scene plane $\pi$ projected onto the image plane. What I ...
2
votes
2answers
33 views
Use Schur to prove that for any norm $\lim_{n \rightarrow \infty} ||A^n|| = 0 \text{ iff } \rho(A) < 1$ where $\rho$ denotes spectral radius.
I know I need to prove both directions of the biconditional, but am having some trouble in both ways.
First, looking left to right, If we assume the limit equals zero, it means that the $A$ matrix ...
0
votes
0answers
17 views
Confusing Proof on Orthogonality and Least Squares
I'm reviewing this textbook proof and am struggling to understand the Least Squares proof that gives us the normal equations. They write:
Suppose $\hat x$ satisfies $A\hat x = \hat b$. By the ...
0
votes
0answers
10 views
Unique decomposition of an affine bijection
Let $E$ be an affine space attached to a $K$-vector space $T$, $a\in
E$ and $u:E\rightarrow E$ an affine bijection. There exists a unique
affine bijection $u_1:E\rightarrow E$ such that $u_1(a)=a$ ...
0
votes
1answer
23 views
Distributive property of matrix-vector multiplication?
I know that matrix multiplication is not distributive but what about matrix-vector multiplication? If $A \in \mathbb{R^{m \times n}}$ and $\vec{x}+\delta \vec{x} \in \mathbb{R^{n \times 1}}$. Then can ...
0
votes
0answers
13 views
How Does One Use a Rotation Matrix to Calculate the Rotation About an Arbitrary Axis?
I have a rotation matrix for a body rotating about some arbitrary axis. I can determine the axis of rotation and the corresponding magnitude of the rotation. But I would like to know how to ...
0
votes
0answers
21 views
Bivectors and (2,0)-tensors
Given a vector space $ V $ of finite dimension $n$, you can consider his dual $ V^* $ i.e. the space of the linear functionals $ \omega:V\longrightarrow K $ (where $K$ in the field of the scalars), or,...
0
votes
0answers
11 views
The logistics of a V to W linear transformation [closed]
If T is a linear transformation from V to W, and w is not in Range(T), does that 2w is also not im Range(T)
0
votes
0answers
35 views
How to solve $x^TAy = d$, where $A$ and $d$ are known, x and y remain unknown.
How to solve $x^TAy = d$, where $A$ and $d$ are known, $x$ and $y$ remain unknown.
$x$ and $y$ are vectors; $A$ is a matrix; $d$ is a scalar.
There are constraints for $x$ and $y$, requiring $x$ and $...
0
votes
1answer
25 views
Consider the equation $2R*x^2 - (2R\cos\frac{B-C}{2})*x+r=0$
$2R*x^2 - (2R\cos\frac{B-C}{2})*x+r=0$
We have the triangle ABC
Where $r = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$
R is the circumradius of the triangle ABC.
I have the following answer ...
1
vote
1answer
20 views
Functional Expectation
Problem: Let $X$ be distributed over the set $\Bbb N$ of non-negative integers, with pmf: $$P(X=i)=\frac{\alpha}{2^i}$$
$\alpha$
$E[X]$
For $Y=X$ mod $3$, find:
$P(Y=1)$
$E[Y]$
I will assume $\...
0
votes
0answers
17 views
If $x_1,…,x_{n-1}$ make angle $\le\theta$ with some $n-1$ plane $P$, what is an upper bound for $\angle(P,\operatorname{span}\{x_1,…,x_{n-1}\})$?
It looks like the angle between $P$ and $\operatorname{span}\{x_1,\ldots,x_{n-1}\} := \tilde{P}$ should be at most $\theta$ or possibly some constant multiple. Here $\theta \in (0,\pi/2].$
Tried ...
1
vote
2answers
32 views
Linear independence of solutions to Floquet equation
I have a question about solutions to the Floquet equation, i.e.:
$$
\dot{x} = A(t) x
$$
where $x$ is an $N$-dimensional time-dependent column vector and $A(t)$ is a $T$-periodic $N \times N$ matrix.
...
1
vote
1answer
36 views
How to prove that the general solution of a homogeneous linear first order ODE has dimension 1?
I would like to prove it by using Linear Algebra.
I first defined a Linear Application on an open interval $I$
$$L(.) : C^1(I) \to C(I).$$
$$y(x) \to y(x)'+p(x)y(x)$$
For a Homogeneous Equation $L(y)=...
0
votes
1answer
46 views
Difference between $p\left(x\right)\cdot{q\left(x\right)}$ and $\left(p\cdot{q}\right)\left(x\right)$
Let $p\left(x\right),q\left(x\right)\in{R\left[x\right]}$ such that $p\left(x\right)=x$ and $q\left(x\right)=x+a$.
My lecture notes state that "$p\left(x\right)\cdot{q\left(x\right)}=x^2+xa$ and $...
0
votes
0answers
21 views
How to calculate determinant? [duplicate]
How to calculate the following determinant:
$\begin{vmatrix}
a & b & b & ... & b \\
b & a & b & ... & b \\
. & . & . & . ... & .\\
b & b &...
0
votes
1answer
19 views
Proving that the matrix representation of a nilpotent linear operator is upper-triangular with diagonal entries $ = 0$
Let T be a nilpotent operator on an $n$-dimensional vector space $V$, and suppose that $p$ is the smallest positive integer for which $T^P=T_0$ (zero-transformation).
I have so far proved the ...
0
votes
1answer
19 views
Finding the eigenvalues and an orthornormal basis of eigenvectors of an orthogonal mapping
Let $\alpha = \{\vec{a_1},\vec{a_2},\vec{a_3},\vec{a_4}\}$ be an orthonormal basis for $\mathbf{R}^4$. Let $\mathbf{A}:\mathbf{R}^4 \rightarrow \mathbf{R}^4$ be a linear orthogonal map with the ...
0
votes
0answers
32 views
Nonlinear Numbers in Linear Algebra
I am writing an electronic circuit simulator, I know there are already tons of them out there but writing one yourself deepens your understanding tremendously.
My approach that works for linear ...
0
votes
1answer
39 views
If two vectors are orthogonal then $\|a + b\|^2=\|a\|^2 + \|b\|^2$
Can we say that if $a$ and $b$ are orthogonal (meaning $\langle a,b \rangle = 0$) then
$\|a + b\|^2=\|a\|^2 + \|b\|^2$
must be correct?
Reasoning:
$\|a+b\|^2 = \langle{a+b,a+b}\rangle \underset{...
0
votes
0answers
16 views
Morphism of complete collineation
A complete collineation of depth $d$ is the following data:
$2(d+1)$ vector spaces $V_i$ and $W_i$ (of dimension d) and $d+1$ linear maps $\lambda_i:V_{i} \longrightarrow W_{i}$ for $i\in {0,...,d}$ ...
0
votes
0answers
18 views
$v_1,…,v_m,w$ linearly independent $\iff$ $w \notin \text{span}(v_1,…,v_m)$
Let $v_1,...,v_m$ linearly independant in $V$ and $w\in V$. Show that $v_1,...,v_m,w$ is linearly independent $\iff$ $w \notin \text{span}(v_1,...,v_m)$. It might be a "classic" result in ...
2
votes
0answers
18 views
Comparing GMRES to equivalent algorithms such as GCR
If I understood correctly both GMRES and GCR minimise the residual norm on $x_0+K_k$. Hence, they produce the same approximations over the iterations. The difference comes from the fact that GMRES ...
