Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, Hamel basis, dimension, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, etc. For questions specifically concerning ...
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14 views
Quadratic Functions
Consider the strictly convex quadratic function $f(x) = \frac{1}{2}x^tPx - q^tx + r,$ where $P \in \mathbb{R}^{n \times n}$ is a positive definite matrix, $q \in \mathbb{R}^n$ and $r \in \mathbb{R}.$ ...
1
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1answer
15 views
Let $v_1 = (1, 0); v_2 = (1,-1) \space\text{and} \space v_3 = (0, 1).$
I am stuck on the following problem :
Let $v_1 = (1, 0); v_2 = (1,-1) \space\text{and} \space v_3 = (0, 1).$ How many linear transformations
$T \colon \Bbb R^2 \to \Bbb R^2$ are there such ...
1
vote
1answer
16 views
What is the fastest algorithm to solve the eigenvector of a transition matrix of a Markov Chain?
Given a transition matrix of a Markov chain, $P$, I want to solve the left eigenvector of $P$, namely a row vector $\alpha$ such that
$$
\alpha P = \alpha
$$
I know the algorithm to solve a linear ...
2
votes
0answers
37 views
Identities with Adjoints
The classical adjoint $\operatorname{adj}(A)$ of a square matrix $A$ has its $(i,j)$-th entry equal to the $(j,i)$-th cofactor (signed minor) of $A$. If $\det(A)\neq0$ we can define the inverse ...
2
votes
4answers
55 views
Diagonalizable matrices in $M_{2\times 2}(\mathbb{F}_2)$
List all diagonalizable $2\times 2$ matrices over the a field $F$ consisting of two elements $0$ and $1$.
I want to try and do this using C++, but perhaps this isn't the place to ask. I have an idea ...
0
votes
0answers
17 views
Finding solutions to the equation
I want to find possible solution satisfying both the equation:
$\sum_{i=1}^{n} f_i^{2} = n$
$\sum_{i=1}^{n} f_i=0$
As the number of equations less than number of variables can we just comment on ...
1
vote
1answer
26 views
Sum of unitary transformation
I am having struggle with this question.
suppose I have two unitary matrices.
Is their sum is normal ?
I am try to give an example to show it is not true and I can not find.
I try to proof and I ...
4
votes
3answers
513 views
Cayley-Hamilton theorem on square matrices
Can anyone help me by giving the proof of the Cayley-Hamilton theorem? It states that every square matrix $A$ satisfies its own characteristic equation:
$p_{A}(A)=0$.
I could prove it when $A$ has ...
0
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0answers
14 views
what are the structures of linear preservers of right matrix majorization?
what are the structures of linear preservers of right matrix majorization?
I think the linear preserver $T:\mathbb{R_p}\rightarrow \mathbb{R_n}$ should be of the form $T(x)=rxA$, where r is a scalar ...
2
votes
1answer
37 views
Questions on differential equations of matrices
I have a differential equation $$N'_x(x)=G(x)N(x)$$ where $N, G$ are $2\times2$ matrices depending on $x$, and $G$ satisfies $\sigma G+G\sigma=0$, $\sigma$ is one half of the pauli matrix, i.e. ...
0
votes
2answers
27 views
Non-parallel vectors confusion
I've got a section in my textbook about non-parallel vectors, it says:
For two non-parallel vectors a and b, if $\lambda a + \mu b = \alpha a + \beta b$
then $\lambda = \alpha $ and $\mu = \beta $
...
3
votes
3answers
119 views
Let $\alpha$ and $\beta$ be two distinct eigenvalues of $A$ then $ A^3 = \frac{\alpha^3-\beta^3}{\alpha-\beta}A-\alpha\beta(\alpha+\beta)I$?
Let $\alpha$ and $\beta$ be two distinct eigenvalues of a $2\times2$ matrix $A$. Then which of the following statements must be true.
1 - $A^n$ is not a scalar multiple of identity matrix for any ...
0
votes
1answer
14 views
Existence criteria for the LU decomposition of a tridiagonal matrix
In this link, the following result is presented without proof:
Let $a, b, c$ be the lower off diagonal, diagonal, and upper off diagonal elements of a tridiagonal matrix. A pivotless LU ...
0
votes
3answers
48 views
Diagonalizable Operators: An Operational Extension
Let $T$ be a diagonalizable operator on a vector space $V$. Prove that the operator
$$a_nT^n + a_{n-1}T^{n-1}+\cdots+a_1T+a_0 Id_V$$
on $V$ is also diagonalizable for any scalars $a_1, ...
1
vote
1answer
43 views
Elementary Linear Algebra
What does it mean when someone says "find a fundamental set of solutions for the system y' $=A$ y"?
That is, the system
$$ {\bf{y'}} =A {\bf{y}}. $$
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0answers
24 views
Orthogonalization of a set of random vectors
Suppose $w_1$ and $w_2$ are zero-mean jointly Gaussian random vectors. Further suppose that they have a covariance matrix given by
$$
\mathbf{cov}\begin{bmatrix}w_1 \\ w_2\end{bmatrix}
= ...
2
votes
3answers
29 views
linear Transformation of polynomial with degrees less than or equal to 2
I would like to determine if the following map $T$ is a linear transformation:
\begin{align*}
T: P_{2} &\to P_{2}\\
A_{0} + A_{1}x + A_{2}x^{2} &\mapsto A_{0} + A_{1}(x+1) + A_{2}(x+1)^{2}
...
0
votes
0answers
41 views
How to estimate linear operator?
If I have an input column vector $\mathbf{x}$ with length $N$ that is linearly transformed by an $N \times N$ matrix $\mathbf{T}$ into $\mathbf{y}$:
\begin{align}
\mathbf{y} = \mathbf{T}{x}
...
1
vote
6answers
10k views
Finding a unit vector perpendicular to another vector
For example we have the vector $8i + 4j - 6k$, how can we find a unit vector perpendicular to this vector?
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0answers
21 views
Proving that this is not a positive operator
Let $\rho$ be a density operator (i.e., it is an ortho projection with rank one, and also a positive operator).
Say $X = X^*$ with a spectral decomposition $X = 1P_1 + 4P_4 + 16P_{16}$, and $Y = Y^*$ ...
0
votes
0answers
18 views
Conjugacy classes for su(2)
I am wondering how to calculate the conjugacy classes of the Lie algebra su(2). My guess is that they can be easily evaluated under the similarity transformations but I am not sure it that is all to ...
0
votes
1answer
55 views
Definition of the inverse matrix and matrix of minors
Explain what it means that a matrix $A$ is invertible. Define the inverse matrix $A^{-1}$.
I said:
A matrix $A$ is invertible if $\det(A) \neq 0$. The inverse matrix $A^{-1}$ is a matrix such ...
1
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0answers
55 views
minimization of function $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$?
I have the following questions referring to this link to a previous question on this site : Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
a) Explain why ...
3
votes
3answers
442 views
Linear Independent Rows vs. Columns
$A$ is an $M\times N$ matrix with linearly independent rows and linearly independent columns. Prove that $A$ must be square matrix.
0
votes
1answer
32 views
Dimension of the set of self-adjoint operators
I'm trying to figure out what the dimension of the set of self-adjoint operators on V would be, or in more concrete terms:
Let $dim V =n$. Let $S(V)$ denote the set of self-adjoint linear operators ...
0
votes
1answer
39 views
Easy way to check for a valid solution in this triple equality?
Let's say I have the following equalities
$a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 = b_1x_1 + b_2x_2 + b_3x_3 + b_4x_4 = c_1x_1 + c_2x_2 + c_3x_3 + c_4x_4$
Where the $a$'s, $b$'s, and $c$'s are known, ...
1
vote
4answers
636 views
Positive Definite Matrix Determinant
Prove that a positive definite matrix has positive determinant and
positive trace.
In order to be a positive determinant the matrix must be regular and have pivots that are positive which is ...
0
votes
0answers
12 views
Frequency determination
the time averaged total energy, $\bar E$,
has the following $\varepsilon$ expansion in $D$ dimension:
\begin{equation}
\bar{E}=\varepsilon^{2-D}\frac{E_0}{2\lambda}+ \varepsilon^{4-D}E_1
...
3
votes
1answer
88 views
Arrangements of affine hyperplanes
Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Call a function $f:X\longrightarrow \mathbb{R}$ linear if it is of the form
$$
f(\bar{x})=a_1x_1+\ldots+a_nx_n+b
$$
for some $a_i,b\in\mathbb{R}$.
Now ...
2
votes
1answer
159 views
Diagonalizing a Unitary Matrix
I'm trying to diagonalize the following unitary matrix:
$\frac {1}{\sqrt{5}}\begin{pmatrix} 1 &2 \\ 2i &-i
\end{pmatrix}$
My approach is to find the eigenvalues and eigenvectors in the usual ...
0
votes
1answer
47 views
How to show this matrix is invertible?
Let $f:H \times H \to \mathbb{R}$ be a mapping with $H$ a Hilbert space.
Let $A$ be a matrix with entries $a_{ij}=f(b_i, b_j)$ with
$$a_{ii}=f(b_i, b_i) \geq C\lVert b_i\rVert_{H}^2.$$
Suppose $b_i ...
9
votes
2answers
256 views
Fast computation/estimation of the nuclear norm of a matrix
The nuclear norm of a matrix is defined as the sum of its singular values, as given by the Singular Value Decomposition of the matrix itself. It is of central importance in Signal Processing and ...
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0answers
34 views
Can anyone help me with this algebra problem?
Let $(\lambda_i)_{i = 1}^n$ be a sequence of real numbers.
For each $1\le i\le n$, define $u_i$ as
$$u_{i}:=1-\omega\cdot(1-\lambda_{i})$$
where $\omega$ is a non zero real number.
Show that we can ...
6
votes
2answers
66 views
Two terms that I want to understand: weakest topology and jointly continuous (in the following context).
I was reading an article online, please help me to understand the following lines (in bold letters). -
Topological structure:
If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and ...
1
vote
1answer
53 views
Projection and inner product space
Definition: Let $V$ be vector space, and $U$, $W$ be two subspaces such that $V=U\oplus W$.
We know that there exists for each $v \in V$ only one $u \in U$ and only one $w \in W$ such that $v=u+w$. ...
4
votes
3answers
87 views
How to find 3 x 3 matrix inverses
Is there a way of finding the inverse of a $3 \times 3$ matrix without forming an augmented matrix with the identity matrix? Also, is there a quick way of checking that a $3 \times 3$ matrix's ...
3
votes
1answer
69 views
Property of the trace of matrices
Let $A(x,t),B(x,t)$ be matrix-valued functions that are independent of $\xi=x-t$ and satisfy $$A_t-B_x+AB-BA=0$$ where $X_q\equiv \frac{\partial X}{\partial q}$.
Why does it then follow that ...
0
votes
0answers
21 views
Proof of eigenvectors of a rotation matrix in complex plane
In Linear algebra, how does one find the eigenvectors of a rotation matrix above the complex vector space.
Given the following matrix
...
2
votes
1answer
25 views
How to Find the Center of a Parallelogram
I want to find the center of a parallelogram in order to use it in my java program. I have four coordinates of the parallelogram and I want to find the center coordinate of the parallelogram. It seems ...
3
votes
0answers
47 views
Eigenbasis of a Hilbert space: isomorphism
Let $K$ be a matrix containing the dot product between points in a Hilbert space $\mathcal{H}$ (assume that it is finite-dimensional). Then, we could form a basis using the eigenvectors of a normal ...
1
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0answers
24 views
Some remarks on JCF which I want to get verified
Jordan Cannonical Form is applicable for all the linear operators whose characteristic polynomial factors linearly over the base field.
Jordan Cannonical Form: Let $T:V_F\to V_F~(\dim ...
1
vote
0answers
33 views
Proof is needed for a lower bound of the maximal eigen-value of a non-negative, irreducible, integer matrix
$A$ is a non-negative, integer, irreducible, $m$ by $m$ matrix. It is well known (Perron-Frobenius) that $A$ has a positive eigen value (denote it by $\lambda$) with a positive eigen vector ($x$). It ...
2
votes
1answer
34 views
How to frame this set of linear equations?
I have the following set of equations, as an example
$2x + 1y + 2z = A$
$0x + 2y + 2z = A$
$1x + 2y + 1z = A$
I assume this can be rewritten as a matrix? How can I check if a solution exists such ...
2
votes
1answer
79 views
Finding Constrained Subsets of Parameters in Larger Poorly-Constrained System of Linear Equations
I have a system of linear equations. The system is not well-constrained (I have more parameters than independent equations). What is the easiest way to identify the subset of parameters that are ...
2
votes
1answer
198 views
Change of coordinate system on a sphere
This might take a while to explain, so bear with me:
I've got a perfect sphere. I've set up an arbitrary longitude/latitude ("angle") coordinate system on it (imagine an equator around the middle, ...
6
votes
2answers
121 views
does a matrix like this exist?
Question:
Does a matrix $A \in M_{3 \times 3}(F)$ exist s.t. $A^4=
\begin{bmatrix} 0&0&1\\0&0&0\\0&0&0\end{bmatrix}$
What I thought:
I think it doesn't. How do you start a ...
1
vote
1answer
37 views
Special linear transformations
Special linear transformations are matrices with determinant equal to 1.
What additional properties do such transformations have compared to "regular" linear transformations?
2
votes
1answer
48 views
If we know the eigenvalues of a matrix $A$, and the minimal polynom $m_t(a)$, how do we find the Jordan form of $A$?
We have just learned the Jordan Form of a matrix, and I have to admit that I did not understand the algorithm.
Given $A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 2 & 1 & -1 ...
1
vote
1answer
43 views
Find the transform
I have the paper with 3 points on it. I have also a photo of this paper. How can I determine where is the paper on the photo, if I know just the positions of these points? And are 3 points enough?
It ...
16
votes
1answer
151 views
Properties of the Cone of Positive Semidefinite Matrices
The set of positive semidefinite symmetric real matrices form a cone. We can define an order over the set of matrices by saying $X\geq Y$ if and only if $X-Y$ is positive semidefinite. I suspect that ...
