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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.

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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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: $$ \text{Trace}(A^{\top}A ) \ge \rho(A)?$$
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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
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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
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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
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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
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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
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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
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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 ...