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 ...
0
votes
1answer
15 views
Given 5 matrices in $M_2(\mathbb{R})$ is this algebra allowed?
Let $A,B,C,D,E \in M_2(\mathbb{R})$
I'm asked to prove or disprove that if the set $A = \{EA,EB,EC,ED\}$ is linearly independent so the set $\{A,B,C,D\}$ is linearly independent.
I was having ...
0
votes
0answers
16 views
Confusions about a factor arising in certain equations
We have known some solitary wave solutions, given by(equations 1 to 5)
$$
\phi_1=p_1\cos \tau \tag{1}$$
$$\phi_2=\frac16 g_2p_1^2\left(\cos(2\tau)-3\right)\tag{2}$$
$$\phi_3=p_3\cos ...
3
votes
1answer
26 views
How to show $SL_{n}(\mathbb{R})=\bigsqcup_{w\in W}LwU$ where L (or U) are lower(or upper) triangular matrix?
I'd like to ask a homework problem that causes me many troubles for days. The problem is like below :
Let W denote the subgroup of permutation matrices in $SL_{n}(\mathbb{R})$. Show the following ...
0
votes
0answers
29 views
Characteristic Polynomial by Induction
I like know if I will be able to prove by induction, that the characteristic polynomial of the matrix $U_{ij} = (-1)^{\delta{jx_0}}(\dfrac{2}{N}-\delta_{ij})$ is ...
3
votes
2answers
63 views
Does exists a matrix $B$ such that $A^TA=A^TB+B^TA$? with $B^TB$ being a diagonal matrix and $A$ an incidence matrix
$A$ is a incidence matrix for some undirected graph.
$A^TA$ is a positive definite matrix, so I know that we can factorize it as $A^TA = C + C^T$
There exists always a matrix $C$ such that $C = ...
0
votes
1answer
27 views
Bilinear form matrix coordinates
I don't know how to solve this problem - I have to transform the coordinates of a bilinear form that has matrix
$$
\begin{pmatrix}
1 & 2 & -2 \\
2 & -2 & 3 \\
...
2
votes
1answer
43 views
Linear Algebra, Parseval's Identity
How does one go about proving Parseval's identity?
Let ${v_1, v_2, ..., v_n}$ be an orthonormal basis for a a finite-dimensional inner product space $V$ over some field $F$. For any $x, y$ in $V$, ...
2
votes
0answers
46 views
If $f$ is an endomorphism satisfying $f^2 = f^3$, it can have finite invariant straight lines. [closed]
Let $f:\mathbb R^3 \to \mathbb R^3$ be an endomorphism satisfying
$$
f^3 = f^2 \neq 0,
$$
then it can have a finite number of invariant straight lines.
If we suppose that $A$ has a finite ...
3
votes
3answers
61 views
What is the relation between vectors in physics and algebra?
Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. ...
1
vote
1answer
48 views
Eigenvalues and eigenvectors of AB and BA, proof.
$A$ is an $n \times k$ matrix and $B$ is an $k \times n$ matrix.
If $v_1, ..., v_l$ are linearly independent eigenvectors of $BA$ corresponding to a single nonzero eigenvalue $c$, then $Av_1, ..., ...
3
votes
3answers
40 views
How to prove that sequence spaces $l^{p}, l^{\infty}$ and function space $C[a. b]$ are of infinite dimension
I am studying about the sequence space $l^{p}, l^{\infty}$ and function space $C[a. b]$. It is mentioned in the book that all of these spaces are of infinite dimension. I want to prove that these ...
0
votes
1answer
25 views
The polynomial subspace
Let $A$ be a set of 6 polynomials in $\mathbb{R}_5[x]$ over $\mathbb{R}$ field,
assume $sp(A) = \mathbb{R}_5[x]$ which of the following is true?
1. It might be that $A$ holds exactly 4 polynomials ...
0
votes
1answer
30 views
Is there a dot product with which the following linear operator becomes Hermitian
Given the linear operator
$A \in L(M_2(\mathbb{C}))$
$A \begin{bmatrix}a & b \\ c & d \end{bmatrix}=\begin{bmatrix}a-b & -a+b \\ d & -c \end{bmatrix}$
Is there a dot product where ...
0
votes
0answers
12 views
Question regarding Iteratively reweighted least squares?
If we have a set of data and then we want to find Iteratively reweighted least squares we know we have to use a weighting function. But I'm not sure how to find that weight corresponding to the data. ...
0
votes
2answers
38 views
find matrix such that $ Ax=(1,1,1)^t$ has exactly three distinct solutions
Does there exist a matrix $3\times 3$ order such that $ Ax=(1,1,1)^t$ has exactly three distinct solutions? If so, find $A$.
I have no idea in this question please help.
0
votes
0answers
32 views
to find the eigenvalues and eigenvectors from linear transformation
Find the eigenvalues and eigenvectors of the linear transformation $T$:$R^3\to R^3$ defined by $T(x_1,x_2,x_3)=(x_1,x_2,x_3)$?
Please tell me how to find the matrix and then I can find the ...
1
vote
0answers
29 views
Diagonalizing/eigenvalues of a particular infinite dimensional matrix
I have trying to show that the continuum limit of n quantum harmonic oscillators gives rise the the klein-gordon field. However, instead of a usual finite string, I want to do it on a ring.
Assume $n ...
2
votes
0answers
28 views
Find the eigenvalues and eigenvectors of an integral equation
I need to find the eigenvalues e eigenvectors of this integral.
A)
$$\int_{0}^{1}(cos^2(x+y)+1/2)\phi (y)dy$$
B)
$$\int_{0}^{1} K(x,y)\phi (y)dy,$$
where
$K(x,y)=x(1-y),\; 0 \le x\le y \le 1$
...
0
votes
0answers
21 views
Solve integral equation of second kind using Fredholm method
I need to solve this integral equation
$$\phi (x)=(x^2-x^4)+ \lambda \int_{-1}^{1}(x^4+5x^3y)\phi (x)dy$$
Using the Fredholm theory of the intergalactic equations of second king.
I really don't ...
1
vote
2answers
28 views
Given a spanning set, what is the span of the 'transpose' of the set?
Given $$sp\left \{
\begin{pmatrix}
a_1\\
a_2\\
a_3
\end{pmatrix}
,\begin{pmatrix}
b_1\\
b_2\\
b_3
\end{pmatrix}
,\begin{pmatrix}
c_1\\
c_2\\
c_3
\end{pmatrix}
\right \} = \mathbb{R}^3$$
What ...
0
votes
1answer
19 views
Schur decomposition of an $n-$by$-n$ matrix
$(\lambda, x)$ is a simple (with multiplicity 1) eigenpair of $A\in \mathbb C_n$ with $x^Hx=1$, $H$ denotes Hermitian.
Use Schur decomposition to show that there exists a nonsingular matrix $(x\ \ ...
2
votes
5answers
76 views
Finding the determinant of $2A+A^{-1}-I$ given the eigenvalues of $A$
Let $A$ be a $2\times 2$ matrix whose eigenvalues are $1$ and $-1$. Find the determinant of $S=2A+A^{-1}-I$.
Here I don't know how to find $A$ if eigenvectors are not given. If eigenvectors are ...
4
votes
1answer
73 views
vector space generated by $\{I,A,A^2,\dots,A^{2n}\}$
$A$ be $n\times n$ matrix then the dimension of vector space generated by $\{I,A,A^2,\dots,A^{2n}\}$ is atmost $n$ right? as $c_0I+c_1A+\dots+c_nA^n=0$ with some nonzero co efficient(cayley ...
0
votes
1answer
27 views
$A\ne 0:V\to V$ be linear,real vec space $V$
$A\ne 0:V\to V$ be linear,real vec space $V$, $\dim V=n$,$V_0=A(V),\dim V_0=k<n$ and for some $\lambda\in\mathbb{R}, A^2=\lambda A$
Then
$\lambda=1$
$|\lambda|^n=1$
$\lambda$ is the only eigen ...
2
votes
2answers
49 views
Let $a_1,\dots,a_n$ be real numbers, and set $a_{ij} = a_ia_j$. [duplicate]
Let $a_1,\dots,a_n$ be real numbers, and set $a_{ij} = a_ia_j$. Consider the $n \times n$ matrix $A=(a_{ij})$.
Then
It is possible to choose $a_1.\dots,a_n$ such that $A$ is non-singular
matrix $A$ ...
0
votes
0answers
21 views
Given that $A,B$ are positive definite matrix [duplicate]
Given that $A,B$ are positive definite matrix, Then I need find which of he following are Positive definite
$A+B$
$AB$
$A^2 +I$
$ABA^{*}$
As $A,B$ are positive definite so $x^TAx>0, x^TBx>0$ ...
1
vote
4answers
83 views
sum of the eigenvalues = trace($A$)?
Is it true that for a square matrix $A$, all of whose eigenvalues exist in the base field, sum of the eigenvalues = trace($A$)?
The result holds in all the matrices I've studied.
1
vote
1answer
21 views
Let $W=\{p(B):p\text{ be a polynomial with real coefiicient}\}$ [duplicate]
Let $W=\{p(B):p\text{ be a polynomial with real coefiicient}\}$ and where $B=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}$ Then the dimension $d$ of the space $W$ satisfies
...
1
vote
2answers
62 views
Abstract Geometry?
Are there similar terms in other areas for the idea the "angle" conveys in geometry ? I find that functions for abstract things such as pressure,electrical currents ( nothing geometric there ) on AC ...
2
votes
1answer
38 views
$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$
$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$, consider the linear map $T:M_2(\mathbb{R})\to M_2(\mathbb{R}):=B\to AB$ Then which of the following are true?
$T$ is ...
2
votes
1answer
60 views
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
Diagonalizable
Positive semidefinite
$0,3$ are only eigenvalues of $J$
Is positive definite
$J$ has minimal polynomial $x(x-3)=0$ so 1, ...
1
vote
2answers
43 views
Consider a matrix with integer entries such that $a_{ii}=1$ and $a_{ij}=0$ for $i>j$ [duplicate]
Consider a matrix with integer entries such that $a_{ii}=1$ and $a_{ij}=0$ for $i>j$
Then
$A^{-1}$ exists and it has integers entries.
$A^{-1}$ exists and it has some entries which are not ...
5
votes
3answers
77 views
In a matrix ring, no zero divisors may have an inverse
In a general ring with 1, a right (left) zero divisor cannot have a right (left) inverse. In a matrix ring over a field, a stronger condition is satisfied: a (right or left) zero divisor cannot have a ...
-3
votes
1answer
51 views
Fantastic Determinant (all $b$ plus multiple of $I$) [duplicate]
$$f(a,b)=\operatorname{det}~\begin{pmatrix} a & b & b & \cdots & b \\ b & a & b &\cdots & b\\ b & b & a &\cdots & b\\ \vdots & \vdots & \vdots ...
2
votes
1answer
33 views
Same linear transformation, different basis.
Let $\beta=\{(1,0,0),(0,1,0),(0,0,1\}$be a basis of $\mathbb{R^3}$ and $g: \mathbb{R^3} \to \mathbb {R^3}$ a linear transformation, which matrix is:
$$G=\begin{bmatrix}1 & 0 &-1 \\ 6 ...
0
votes
0answers
14 views
Given a large non-square linear operator and it's adjoint, how to find the most correlated dimensions
If I have a large blackbox linear operator function F that maps R^n to R^m, and I also have its adjoint function F'; Is it possible to detect highly correlated dimensions with relatively few calls to ...
6
votes
2answers
61 views
$V=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\}$ is isomorphic to $\text{span}\{AB-BA:A,B\in V\}$
Background: Let
$$V:=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\}$$
be the vector space of $3\times 3$ real matrices with vanishing trace, and let $[\cdot,\cdot]:V\times V\to V$ be defined by
...
0
votes
1answer
51 views
Prove: symmetric positive matrix multiplied by skew symmetric matrix equals 0
My teacher gave me this task as preparation for the exam but I'm stuck and not sure if it's true anymore.
0
votes
0answers
53 views
to find dimension of $(\ker f)^\perp$
Let $V$ be an $n$-dimensional inner product space.Let $f:V\to R$ be a linear form. Find the dimension of $(\ker f)^\perp$
0
votes
2answers
44 views
How to verify the orthogonal projection formula?
Let $B = \{\vec{b}_1, \vec{b}_2, \vec{b}_3\}$ a orthogonal basis $\in V^3$.
Verify that $\forall\,\vec{u} \in V^3$,
$$\vec{u} = ...
1
vote
0answers
23 views
$\ast$-homomorphism
Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
0
votes
3answers
63 views
Is the following set empty?
$$
sp\left \{
\begin{pmatrix}
1 \\
-1 \\
1 \\
-1
\end{pmatrix} , \begin{pmatrix}
4\\
-2 \\
4 \\
-2
\end{pmatrix} , \begin{pmatrix}
1\\
1\\
1\\
1
\end{pmatrix}
\right \} \bigcap \left \{ ...
1
vote
1answer
27 views
Symmetric Matrices Using Pythagorean Triples
Find symmetric matrices A =$\begin{pmatrix} a &b \\ c&d
\end{pmatrix}$ such that $A^{2}=I_{2}$.
Alright, so I've posed this problem earlier but my question is in regard to this ...
1
vote
2answers
17 views
Determinant of product of symplectic matrices
In optical ray tracing it's possible to use symplectic matrices. I have a problem with them.
If a matrix $M$ is symplectic, this means that for $M$ the following equation hols:
$$M^T\Omega M=\Omega$$
...
2
votes
1answer
22 views
self-adjoint and eigenvalues properties
I wondering about something.
Let $V$ be an inner product space
$T\colon V\to V$ is a linear map
$T$ is self-adjoint and all the eigenvalues of $T$ are not negative
I need to proof that for all $v$ ...
2
votes
1answer
44 views
If $A$ is positive definite, then $B^TA^{-1}B$ is also positive/negative (semi) definite?
probably this is a basic question, but I can not see it clearly.
If $A \succ 0$, then $B^TA^{-1}B$ is also positive/negative (semi) definite? or in general is undefined?
In addition, you can assume ...
4
votes
3answers
103 views
Where to start when learning math (again)?
I have a few questions I hope you can help me answer.
First, I'll introduce myself. I'm a finance undergraduate student in Australia, but I'm originally from Norway. Throughout school I always loved ...
1
vote
1answer
38 views
Grover Algorithm Orthogonal vectors
I'm study the Grover algorithm. About this picture my lecture say that the expression $|s'\rangle$ in the computational basis is $$|s'\rangle = \dfrac{1}{\sqrt{N-1}}\sum_{x\neq w}|x\rangle ...
1
vote
1answer
60 views
Does such $A,B$ exist?
true/false test: there're $n\times n$ matrices $A,B$ with real entries such that $(I-(AB-BA))^n=0$
I'm cluesless to begin.
1
vote
2answers
80 views
Upper and Lower Triangular Matrices
Given the matrix A=$ \left( \begin{array}{ccc}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8\\
1 & -1 & 2 & 3 \\
2 & 1 & 1 &2\end{array} \right) $, write it in the ...
