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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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Projection of a circle via Matlab

I am currently stuck at this exercise - I dont know where to begin, which forumals should I use :/ You have to implement the projection of a circle as viewed by a camera in 3D with different angles ...
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0answers
9 views

Standard form for the Characteristic Matrix/Polynomial

I'm currently taking Linear Algebra and Differential Equations, and in talking about eigenvalues of a matrix, both professors have given the same information: for some square n x n matrix A, the ...
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0answers
15 views

Derivative of matrix as a function of a vector w.r.t a vector

I want to compute the derivative of the matrix $ diag(x)M $ with respect to $ x $, where $ x \in \mathcal{C}^{n \times 1} $ and $ M \in \mathcal{C}^{n \times m} $. This is how I have approached it, ...
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1answer
25 views

How do the values of $a$ affect definiteness of this matrix $\textbf{A}$?

Let $$\textbf{A} = \begin{bmatrix} -2 & 0 & 1 \\ 0 & -2 & a \\ 1 & a & -2\end{bmatrix}$$ Given that one of its eigenvalues is equal to $-2$, how does its definiteness vary ...
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0answers
18 views

How to show for positive semi-definite, there exist $x \in \mathbb{R}^n$ such that $Bx+c = 0$ and and $\|x\| \leq \Delta$?

Let $B \in \mathbb{R}^{n \times n}$ be symmetric and positive semi-definite such that $B = U\Lambda U^T$, where $U = [u_1,\cdots,u_n]^T$ is an orthogonal matrix with $u_i \in \mathbb{R}^n$, and $\...
2
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1answer
48 views

When can you uniquely determine a square matrix if you know the sum of its rows and columns?

There is a square matrix consisted of only $0, 1, i$. You know the sum of all the numbers in each row and each column. When can you uniquely determine the matrix? Edit: For clarification, there is a ...
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1answer
19 views

Simultaneous Diagonalization With Non-Similar Eigenvectors

So I've been given two diagonal matrices with non matching eigenvectors, A:$$ \begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ -1 & 0 & 1 \\ \end{matrix} $$ and B: $$ ...
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2answers
21 views

Find orthogonal basis for 3x3 Symmetric Matrix

Given a 3x3 symmetric matrix $A$ =\begin{pmatrix} 1 & -1 & -1 \\ -1 & 1 & 1 \\ -1 &1& 1 \end{pmatrix} Find the orthogonal basis corresponding to the above matrix $A$ The ...
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Generator matrix of Reed-Solomon code [on hold]

Given the [10, 3, 8] Reed-Solomon code over $F_{11}$ with evaluation points $\alpha_i = i$ for $i = 1, 2, ..., 10$. How can I construct, with these information, the generator matrix?
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Generator matrix for binary Goppa code

Let $F_8 = F_2 (\alpha)$, where $\alpha$ is a root of $x^3+x+1$. I'd like to construct the generator matrix for a binary Goppa code $T(L, G)$ with Goppa polynomial $G(x) = x+1 \in F_8[x]$ and L the ...
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1answer
55 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 $A^{1000}$. ...
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1answer
31 views

Matrix and vectors (Find 3 Vectors)

I am having a bad time with this matrix and vector situation, and I think the solution is kind a trick in some part of the computation, but I don't know how to find this: Find 3 vectors (different ...
2
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1answer
28 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/...
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1answer
16 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 & ...
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1answer
34 views

How to derive this complex differentiation?

In the finite-deformation theory, the elastic Cauchy-Green strain $\mathbf{E}_e$ is defined as $\mathbf{E}_e=\frac{1}{2}(\mathbf{F}_e^T \mathbf{F}_e-\mathbf{I})$, where the superscript $T$ denotes ...
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0answers
39 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?
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1answer
53 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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1answer
61 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.
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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}\,\...
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1answer
11 views

Efficient matrix inversion after update when the size of the components changes

I have a matrix of the following form: $$K = \begin{pmatrix}A & B \\\ B^{\intercal} & C\end{pmatrix}$$ where $A$ is large compared to $B$ and $C$, and $A$ and $C$ are symmetrical. The $K^{-1}$...
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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/(...
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0answers
14 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 $\mathbb{R^n}$ ...
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1answer
27 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 ...
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2answers
21 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 ...
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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]...
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1answer
21 views

Coordinates of 3D vector in rotated coordinate system (without using a matrix)

The problem: There is a vector with coordinates X,Y,Z. This vector is in a coordinate sytem that has been rotated by A degrees along the X axis and B degrees along the Y axis. I would like to know ...
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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\...
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0answers
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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 $$ \begin{bmatrix} 2 & 1 \\ 0 & 2 \\ \end{bmatrix} $$ ...
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1answer
14 views

About relation in linear polynomial over linear polynomial with matrices.

I was analyzing rational expressions involving linear polynomial over linear polynomial of the form: $${ax+b \over cx+d},\;\text{where $a,b,c,d\in\Bbb R$};$$ Amazingly, these polynomials have some ...
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1answer
11 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-...
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1answer
81 views

Find determinant of a 3-diagonal matrix

Find the value of matrix determinant. $$\begin{vmatrix} x&1&0&0&\cdots\\ -n&x-2&2&0&\cdots\\ 0&-(n-1)&x-4&3&\cdots\\ \vdots&\cdots&\ddots&...
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0answers
32 views

Question of the Cholesky decomposition of symmetric positive definite matrix

This is a exercise on my numerical analysis textbook: Suppose $\mathbf A$ is a positive-definite symmetric matrix, and the Cholesky decomposition is of the form $\mathbf {A} =\mathbf {LL}^{T}$, ...
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0answers
34 views

Eigenvalue of matrix over (Z/pZ)

How can I show for $M\in\text{GL}_d(\mathbb{Z}/p\mathbb{Z})$ with $\text{ord}(M)=p^n$ ($n$ a positive integer), that $1$ is an eigenvalue of $M$? I would be grateful for any hint or advice. Thank ...
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26 views

Linear Algebra : Eigen value [on hold]

If a square matrix of order 10 has exactly 4 distinct eigen value, then find degree of its minimial polynomial.
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1answer
33 views

Inequality w.r.t determinant of a non-negative definite matrix.

I am reading a paper where the author mentioned the following property without proof. Neither can I prove it nor can I find the proof in various textbooks. For any non-negative definite (i.e. ...
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0answers
34 views

Computing Eigenvalues of a large matrix

Let's say a matrix M is composed of: \begin{bmatrix} A & B \\ C & D \end{bmatrix} where $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{m \times n},$ ...
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2answers
66 views

Why does $B^{-1}(AB)B = BA = B(AB)B^{-1}$?

We have square matrices A and B also B is invertible. Why is $B^{-1}(AB)B$ equivalent to $B(AB)B^{-1}$ so that they're both equal to $BA$? If I do this: $B(AB)B^{-1} = (BA)(BB^{-1}) = BA$ but for ...
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0answers
23 views

question about linear algebra in information theory

im learning about hamming code there is equation $x=uG$ where G is generator matrix i need to find generator matrix from the codeword x and u is message bit suppose code word is $(100110)=(100)G$ ...
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2answers
35 views

A Kronecker Product identity

I want to show the Kronecker Product identity listed on Wikipedia: $$\begin{align} \mathrm{vec}(AXB) =(B^T \otimes A) \mathrm{vec}(X) \\ \tag{1} \end{align}$$ Wikipedia does not cite references for ...
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2answers
218 views

Why intuitively do the quaternions satisfy the mixture of geometric and algebraic properties that they do?

[I completely rewrote the question to see if I could make it clearer. The comments below won't make any sense. In fact, my original question has been answered by Eric Wolfsey, so I may restore it.] ...
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1answer
55 views

Prove that this sequence of continued fractions $\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}},\dots$ tends to $1$.

The Problem: I'll write up a couple more terms: $$\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}}, \frac{20}{19+\frac{18}{17+\frac{16}{15+\frac{14}{13}}}}, \frac{30}{29+\...
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0answers
25 views

Name for diagonal and upper off-diagonal entries

This is in connection with this question Is there a term to call the 'main diagonal entries together with the upper off-main diagonal entries'? When we say 'upper off-main diagonal', it doesn't ...
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0answers
17 views

The nearest matrix over unit ball of matrix 2-norm

Given a matrix $X$, let $\mathbf{proj}(X)=\underset{\|Y\|_2\le 1}{\arg\min} \|X-Y\|_F$. Now the question is to solve $\mathbf{proj}(X)$. Proposition: Suppose $X=U \,\mathbf{diag}(\sigma)\,V^T$ is the ...
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3answers
38 views

When are multiplication on matrices commutative?

According to me multiplication on matrices are commutative only when (i) The given matrices are equal (ii) When the matrices are diagonal matrices and of same order. (iii) When a suitable identity ...
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0answers
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Rank-1 modification of correlation matrix

I inherited a problem at work and have trouble wrapping my head around it. It doesn't help I'm insufficiently sophisticated with linalg. Help with or pointing to appropriate literature will be much ...
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0answers
25 views

Multiplication of matrices in back propagation

I was watching a public available video from Stanford (https://youtu.be/d14TUNcbn1k?t=2720) on the mathematics behind back propagation. They proposed a graph: that was then used as an example of back ...
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0answers
18 views

3x3 matrix Eigenvector values does not match

I'm trying to solve a matrix $Q$ (as shown in the SCREENSHOT) to find its eigenvectors. The solutions are provided in the book directly and I was trying to solve it by hand but I cannot match my ...
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0answers
29 views

Find a basis for the row space, column space, kernel, and image of the following matrix verification

For the following matrix: $$ \begin{bmatrix} 1 & 2 & 1 & 3 \\ 2 & 5 & 5 & 6 \\ 3 & 7 & 6 & 11 \\ 1 & 5 & 10 & 8 \\ \end{bmatrix}...
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1answer
27 views

Find the Jordan Canonical Form that is similar with the idempotent matrix A

Find the Jordan Canonical Form that is similar to the idempotent matrix $A$. I know that since $A=A^2$ then $A(A-I)=0$ so the minimal polynomial is $m_A(\lambda)=\lambda(\lambda-1)$. I also know ...
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2answers
36 views

Given a linear function T determine whether $T(1,1)=1$

Given a linear function $T$ such that $T(1,0) = 1$ and $T(0,1) = 0$ then determine whether is $T(1,1) = 1$ . The given conditions are forming a basis matrix \begin{pmatrix} 1 & 0\\ 0 & 1 \...