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.
83,600 questions
-1
votes
0answers
36 views
How to proceed with following problem of algebra
How to proceed with this problem. I tried to form equation but it cant be solve with that.
$$\begin{align}\dfrac{a^2}{2^2 - 1^2} + \dfrac{b^2}{4^2 - 1^2} + \dfrac{c^2}{6^2 - 1^2} + \dfrac{d^2}{8^2 - ...
0
votes
2answers
20 views
Degenerate eigenvalues and finding normalized eigenvectors
I am trying to find the eigenvalues and the normalized eigenvectors of the matrix:
$$\begin{bmatrix}0 & i & 0\\0 & 0& i\\i & 0 & 0 \end{bmatrix}$$
It is stated that the ...
0
votes
1answer
19 views
linear algebra eigenvalue proof question
Prove that a 2 × 2 matrix with just one eigenvalue (of multiplicity two) is diagonalizable if and only if it is already a diagonal matrix.
So, by multiplicity two, I'm guessing it just means you have ...
2
votes
1answer
34 views
How to use eigenvalues and eigenvectors to compute $A^{1000}$
$$A=
\begin{bmatrix}
0.9 & 0.15 & 0.25 \\
0.075 & 0.8 & 0.25 \\
0.025 & 0.05 & 0.5 \\
\end{bmatrix}
$$
I need to use a Python script to compute $A^{1000}$.
...
0
votes
2answers
21 views
Find the sum of some numbers.
I have a number $a=3145$.At first exercise I need to find how many numbers can be formed with the digits of a. $4!=24$ numbers.My problem is at second exercise where I need to find the sum of these ...
0
votes
0answers
8 views
N-driven Vector Space
Find the no of BASES of Vector Space V (n-dimensional) over a finite field of Order ‘q’
0
votes
1answer
25 views
Nilpotence and module of coefficients
Given a transformation $T\in L(V)$, where $V$ is a vector space of finite dimension, show that $\forall \epsilon >0$, there exists a basis $\mathcal B$ from $V$ such that $[T]_\mathcal B = (a_{ij})...
1
vote
0answers
26 views
A problem with indefinite matrix
Let $A$ be a real symmmetric matrix of the type $n\times n$, positive semidefinite of the rank $k$, where $k<n$. Let
$
K:=\left [ \begin {array}{ll}
A & b\\
b^T & d
\end{array} \right ],
$
...
2
votes
1answer
19 views
If $A \in C^{nxn}$ , $A \ge 0 $ and A is sing., there exists a sequence of matrices $C_k$, that $C_k \ge 0$,$|C_k| = 1$ and trace $AC_k \le 1/k$
Question: Show that if $A \in C^{n \times n}$ , $ A \ge 0 $ and A is singular, then there exists a sequence of matrices $C_k$, $k = 1,2,...$ such that $C_k \ge 0$, det $C_k = 1$ and trace $AC_k \le 1/...
2
votes
1answer
22 views
Dimension and cardinality of vector spaces in Coding Theory
Let $V$ be a vector space over $F$, where $\dim(V) = n$. Then $|V| = |F|^{n}$. This is intuitive for me in abstract settings. Consider the standard basis for $V$, $\{e_1,e_2,...,e_n\}$. There are $n$ ...
1
vote
1answer
31 views
if $ A \in R^{n \times n}$ , $A > 0$ and $ b \in R^n$ then the function $\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$ is convex in $R^n$
Show by direct estimates that if $ A \in R^{n \times n}$ , $A > 0$ and $ b \in R^n$ then the function
$$\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$$ with $x$ is convex on $R^n$.
My ...
1
vote
3answers
27 views
Let A be an $n\times n$ real matrix. Then is it true that every eigenvalue of $A^tA$ is a non negative real number?
Let A be an $n\times n$ real matrix. Then is it true that every eigenvalue of $A^tA$ is a non negative real number?
I know $(A^tA)^t=A^tA$ so we can conclude that $A^tA$ is symmetric matrix. Since ...
0
votes
1answer
14 views
What is the normalized graph matrix if the row-sum of proximity matrix is zero?
Let $S \in \mathbb{R}_{\ge 0}^{n \times n}$ be the proximity (or similarity) matrix of a graph, e.g.
$$
S = \left[ \begin{matrix}
0 & 0.9 & 0.3 \\
0.9 & 0 & 0.4 \\
0.3 & 0.4 & ...
0
votes
0answers
4 views
Goppa code (2-ary) with degree r polynomial
Let $G(x) \in F_{2^m} [x]$ be a degree r polynomial. Let $L = (\alpha_1, ..., \alpha_n) \subset F_{2^m} $ be a set of nonroots of G.
Why is the 2-ary Goppa-code $T(L, G) = T(L, G^2) $ where $T(L, G) ...
2
votes
6answers
604 views
Determinant of a matrix with 2 equal rows
In my linear algebra course, we have just proved that if a matrix A contains 2 equal rows, then det(A)=0.
I understand how the proof works, but could somebody offer a more intuitive explanation of ...
4
votes
1answer
44 views
Is $A^2-B^2$ positive definite too when $A-B,B$ is positive definite?
Denote $A,B\in M_n(\mathbb{R})$
If $A-B,B$ is positive definite, it's easy to see $A^2-B^2$ is symmetric.
Now the question is:
Prove or disprove: $A^2-B^2$ is positive definite.
I have checked ...
0
votes
0answers
16 views
Flexible grid update algorithm
I have a 2D square grid where the edges are line segments and the vertices have moved from their initial positions.
Based on external constraints, the vertices are submitted to further motion, under ...
0
votes
0answers
32 views
Difference between $x^{T}Ay$ and $y^{T}Ax$ when $A$ is not symmetric?
$x^{T}Ay=y^{T}Ax$ if $A$ is symmetric. what is the difference between the two when $A$ is not symmetric? Is the difference negligible under some condition?
1
vote
0answers
45 views
How to find the exact Solution to $ax^2+bx+c= \log(dx)$?
I have an equation
$$
ax^2+bx+c=\log{dx}
$$
How can I solve it?
I know I can use Taylor series to approximately find $x$ but I need an exact solution.
Thanks in Advance
1
vote
0answers
20 views
Similarity to power of sets
Show that if $A\in \mathbb{F}^{n×n}$ such that $\sigma (A)=\lbrace 1\rbrace$, then $A$ is similar to $A^k$ for all positive integer $k$.
Show that the converse is not true.
Any hint will do and is a ...
2
votes
1answer
51 views
Does this inequality hold $\operatorname{Trace}(A^TA) \ge \rho(A)$?
Suppose $A \in M_n(\mathbb R)$ is an arbitrary square matrix and $\rho(A)$ is the spectral radius of $A$. Does this inequality hold:
$$ \text{Trace}(A^{\top}A ) \ge \rho(A)?$$
-2
votes
2answers
40 views
Prove $(A^2-A)^2=0 \wedge (A^3+A^2-2A)=0 \implies A^2=A$ [on hold]
Show that if $(A^2-A)^2=0$ and $A^3+A^2-2A=0$ then $A^2=A$.
Any hint is a great help. Thanks.
0
votes
0answers
12 views
To code for principal component in functional PCA [on hold]
I am trying to get the way to find eigenfunction from eigenvalues. As far as I understand from reading that I have to multiply eigenvalues by something to get principle components in functional PCA. ...
0
votes
1answer
59 views
Proving $(AB)^T=B^TA^T$ [duplicate]
What is the proof of this property of matrices:$$(AB)^T=B^TA^T,$$ where $A$ and $B$ are square matrices and $T$ means transpose.
0
votes
0answers
18 views
How do you create the polynomial basis function given data?
According to Wikipedia:
However, i saw that basis functions applied before linear regression looks like:
Is this saying that, Each observation x has D dimensions. then applying the wikipedia ...
1
vote
0answers
16 views
Questions about the basics of coordinate systems and their basis
Since I think I am missing some basic understanding about coordinate systems and their basis, I would really appreciate your help answering my questions. Also if something isn't written mathematically ...
2
votes
1answer
100 views
Normal Operator $\| T^2\| = \|T\|^2$
Given a complex inner product space X, and an operator $T: X \rightarrow X$ is normal i.e. $T^*T=TT^*$
How can we show $\| T^2\| = \|T\|^2$?
By the definition of operator norm, it follows that ||...
1
vote
0answers
21 views
for arbitrary vector $v,u$, is there the matrix X which satisfy the relation exp$[X]\,v=u$?
Nowadays, I'm studying for exponential map of Lie group.
my question is, To make the form of exp$\begin{pmatrix}x_{11}&x_{12}&\cdots \\x_{21}&\ddots \\ \vdots\end{pmatrix}$,
I have to ...
1
vote
0answers
17 views
Representing a complex line as a directed ellipse
Consider nonzero $v = v_r + iv_i \in \mathbb{C}^n$, It can be thought of as an ordered 2-tuple of vectors $(v_r, v_i)\in \mathbb{R}^n\times\mathbb{R}^n$.
The complex line generated by $v$ is
$$\{r[(\...
0
votes
0answers
12 views
Solving a linear system up to scaling
Problem
Let $v_i \in \mathbb{R}^n$ and $u_i \in \mathbb{R}^m$, where $n \ge m$.
We have $m+1$ pairs $(v_i, u_i), i=1,...,m+1$, where only $m$ many $v_i$ are lineary independent (i.e., $\mathrm{dim}\,\...
0
votes
0answers
22 views
I have this matrix. It is a Gram matrix of some bilinear form on the R^n.
(1 1/2 1/3 1/4 ... 1/(n-1) 1/n)
(1/2 1/3 1/4 1/5 ... 1/n 1/(n+1))
(1/3 1/4 1/5 1/6 ... 1/(n+1) 1/(n+2))
(1/4 1/5 1/6 1/7 ... 1/(n+2) 1/(n+3))
(...................................)
(1/(n-3) 1/(n-2) 1/(...
0
votes
0answers
11 views
Nonnegative matrix exercise by Minc
I must show that:
Let $A$ be a nonnegative $n\times n$ matrix, show that $A$ is reducible iff there exists a proper subset $\{e_{j_1},e_{j_2},...,e_{j_k}\}$ of the standard basis of $\mathbb{R^n}$ ...
1
vote
1answer
45 views
How to simplify this expression $0^x*y^0 + 1^x*y^1 + 2^x*y^2 + 3^x*y^3 +…+n^x*y^n$. [on hold]
How to simplify this expression $$(0^x*y^0) + (1^x*y^1) + (2^x*y^2) + (3^x*y^3) +.....+(n^x*y^n)$$.
0
votes
0answers
22 views
Change of base matrix between displaced and rotated coordinate systems
I have a function that solves a problem when a specific angle equals $0$. The same function can be used with non-zero angles if you compute the problem from other coordinate system.
The scheme of the ...
3
votes
2answers
20 views
Showing a mapping is bijective if and only if a matrix is invertible
Let $\mathbf{A}$ be an $n\times n$ matrix and let $\mathbf{c}$ and
$x_{\star}$ be point in $\mathbb{R}^{n}$. Define the affine mapping
$\mathbf{G} : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ by
...
1
vote
1answer
19 views
Rotation matrix in a non-orthonormal basis.
I have no idea how to approach this sort of problem. I have a non-orthonormal basis $\vec{f_{1}}=\vec{e_{1}}+\vec{e_{2}}$ and $\vec{f_{2}}=\vec{e_{1}}-2\vec{e_{2}}$ where {$\vec{e_{i}}$} is an ...
1
vote
1answer
21 views
Matrixes with common parameters to result in no inverse
I've been given three matrices $A, B \ \& \ C$ which are defined as follows:
$$ A = {
\left[
\begin{array}{ccc}
b & 5 & 8 \\
c & 1 & 3 \\
a & 4 & 3 \\
\end{array}
\right]...
0
votes
1answer
27 views
A linear transform mapping one billinear map to another may not exist (counter-example)
$$\newcommand{\im}{\mathrm{im}\;}\newcommand{\Span}{\mathrm{Span}}\newcommand{\rank}{\mathrm{rank}\;}$$For a bilinear map $\phi : V \times W \to E$ define the first nullspace as
$$
N_1(\phi) = \{ v \...
0
votes
0answers
31 views
XOR resolve using Normal equation
As part of Deeplearning, I was trying to resolve XOR (Exclusive OR) Gate Function using Normal Equation against Least Square Error.
MSE = $$\frac{1}{4}\sum_{x \in X}(f^*(x) - f(x;\theta))^2$$...
1
vote
2answers
40 views
Given $R$ and its eigenvalues, find the eigenvalues of $R + 2I$:
I have a problem solving an exercice, that I expose in the following.
Let $\textbf{R}$ be a $3\times3$ matrix with eigenvalues $\lambda = \{-4,-2,\ 2\}$. What are the eigenvalues of $\textbf{R} + 2\...
0
votes
0answers
10 views
skew-symmetric non-degenerate bilinear space has even dimension [duplicate]
How to prove skew-symmetric non-degenerate bilinear space has even dimension
with skew-symmetric defined as $(x,y) = - (y,x), \forall x,y \in V$, where $V$ is the vector space
1
vote
0answers
46 views
Geometrically, what does it mean for a matrix to be degenerate? (i.e. have non-distinct eigenvalues)
I'm trying to understand matrix operations as geometric transformations. For example, in the $2$x$2$ case, the matrix $$
\begin{bmatrix}
2 & 1 \\
0 & 2 \\
\end{bmatrix}
$$ ...
0
votes
1answer
15 views
How to show that the multivariate normal depends on the data only through $\Sigma x_n$ and $\Sigma x_n x^T$
I've shown something similar for the 1-dimensional case. That $\Sigma x_n$ is the sufficient statistic of the gaussian mean and $\Sigma x_n$,$\Sigma x_n^2$ are the sufficient statistics of the ...
0
votes
2answers
24 views
Show that $x, cosx,$ and $\frac{x^2}{1+x^2}$ are linearly independent in $C(\mathbb{R})$?
I assume by $C(\mathbb{R})$ the question means the vector space of continuous real functions, but I'm not completely sure of that.
How might I go about formally proving this? Obviously I could say ...
0
votes
0answers
21 views
Given $A$ is normal, prove that if $A$ is unitary then, the modulus of eigenvalues of $A$ is precisely $1$.
Claim: Suppose $A \in \mathbb{C}^{n \times n}$ is normal. If $A$ is unitary, then $|\lambda| = 1$ for all $\lambda \in \sigma(A)$.
where $\sigma(A)$ is the spectrum of $A$, i.e. the set of the ...
-1
votes
0answers
15 views
Question about symmetric linear transformation [on hold]
In the real inner product space $V$, a linear transformation $A$ is a symmetric transformation if $\langle A\alpha,\beta\rangle =\langle\alpha,A\beta\rangle$
In my book it says a linear ...
4
votes
2answers
33 views
Finding the characteristic polynomial of the $T: \mathcal{M}_{n} (\mathbb{R}) \to \mathcal{M}_{n} (\mathbb{R})$ given by $T(M)=M^{\text{tr}}$
Here's a problem from Larry Smith's Linear Algebra textbook:
Let $\mathcal{M}_{n} (\mathbb{R})$ be the set of real matrices of
order $n \times n$. Let $T: \mathcal{M}_{n} (\mathbb{R}) \to \...
0
votes
1answer
10 views
How to write the parity-check matrix of Hamming code?
I have read some questions about this topic, but I am still not clear about some concepts about Hamming code.
If we want to write a parity-check matrix for $n$ information positions(with single error-...
0
votes
0answers
19 views
Relationship between two lines
The problem is $x-y-1=0$ and $x-y+1=0$
So far, as finding $k$ for both lines is one, and so it is parallel. But having the rule of $b1 \neq b2$, can it still be a parallel? Because their $b$ is $1$. ...
0
votes
1answer
29 views
Finding smith normal form of $x-A$
$$
A =
\quad
\begin{pmatrix}
1 & 1 &0 &0 \\
-1 & -1 & 0& 0\\
-2&-2 & 2 & 1 \\
1& 1&-1 & 0
\end{pmatrix}
\quad $$
I want to find Smith normal form of ...
