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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.

2
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0answers
7 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/...
0
votes
0answers
9 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 & ...
1
vote
0answers
22 views

Find rank of a matrix over $\mathbb{Z}_7$

Consider the following matrix $$ A = \begin{pmatrix} \overline{3}a & \overline{3} & a+\overline{1} & b + \overline{1} \\ \overline{-3} & \overline{6} & \overline{1} & \overline{...
0
votes
0answers
15 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 ...
-1
votes
0answers
17 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 any condition?
2
votes
1answer
47 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: $$ \text{Trace}(A^{\top}A ) \ge \rho(A)?$$
1
vote
1answer
55 views

Proving $(AB)^T=B^TA^T$

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
11 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
8 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 $K^{-1}$...
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
10 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 $\mathbb{R^n}$ ...
0
votes
0answers
20 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

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
19 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 ...
1
vote
2answers
39 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\...
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
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 ...
0
votes
1answer
9 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-...
3
votes
1answer
51 views

find the bmatrix $\det(x,1,0,0,\cdots)$

This problem comes from an advanced algebra book. He is not allowed to use the Jordan type of processing. He can only use elementary transformation knowledge to solve it. I haven't solved it for a ...
1
vote
0answers
31 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 $\mathbf {A} =\mathbf {LL}^{T}$, ...
0
votes
0answers
31 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 ...
-1
votes
0answers
24 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.
0
votes
1answer
32 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. ...
0
votes
0answers
32 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 $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{m \times n},$ ...
1
vote
2answers
64 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 ...
-1
votes
0answers
21 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 $(100110)=(100)G$ ...
1
vote
2answers
31 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 ...
8
votes
2answers
185 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.] ...
1
vote
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+\...
0
votes
0answers
24 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 ...
0
votes
0answers
16 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 ...
1
vote
3answers
37 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 ...
1
vote
0answers
21 views

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 ...
0
votes
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 ...
-1
votes
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 ...
0
votes
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}...
1
vote
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 ...
0
votes
2answers
35 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 \...
0
votes
0answers
34 views

Why do solving a $4\times4$ matrix using Gaussian elimination have $58.66$ arithmetic operations?

The question reads "Show that the total number of arithmetic (multiplication, divisions and additions) operations needed to solve the matrix below using Gaussian elimination is approximately 58.66". ...
1
vote
1answer
41 views

$A=\lambda I_n\iff (\forall M,N\in M_n(\mathbb{R}),~ MN=A \Rightarrow ~ NM=A)$

here is my question: Let $n\geqslant 1$ and $A\in M_n(\mathbb{R})$. Show that $$\boxed{\exists \lambda\in\mathbb{R}^*,~ A=\lambda I_n\iff (\forall M,N\in M_n(\mathbb{R}),~ MN=A \Rightarrow ~ NM=A)}$...
0
votes
1answer
16 views

Parametrize (as a subset of R5) the solution space of the system of equations

I am just wondering how I would parametrize the system of equations from the augmented matrix I know the following: v = -2 - 4w -3z w = 0 x = 0 y = 8 - 5z z = 0 \begin{bmatrix} 1 & 4 & -2 &...
0
votes
1answer
33 views

Writing $\beta$ and X for a qualitative model

Express the following model in matrix form, ie: specify $\beta$ and $X$ so that the model can be written as $Y = X \beta + \epsilon$. The model $Y_{ij} = \mu_i + \epsilon_i$ where $Y_{ij}$ represents ...
0
votes
0answers
18 views

Simplex method using two phase

(P) minimize: $z=x_1+x_2$ subject to : $$\begin{aligned} x_1 + 2 x_2 &\geq 4 & &\text{Eq.1} \\ 2x_1 + x_2 &\geq 6 & &\text{Eq.2} \\ -x_1 + x_2 &\leq 1 & &...
3
votes
1answer
54 views

Let $A\in M_{n \times k}(\mathbb{R})$. Show that $\det(A^TA)=\sum \det(B^2)$ where the sum runs through all $k \times k$ submatrices $B$ of $A$.

Let $A\in M_{n \times k}(\mathbb{R})$. Show that \begin{equation} \det(A^TA)=\sum \det(B^2) \;\;\;\;\;\;\;(*) \end{equation} where the summation runs through all $k \times k$ submatrices $B$ of $A$ ...
0
votes
2answers
33 views

Numerically verify a matrix is positive semidefinite

I am trying to numerically (in Julia) verify that A symmetric matrix $\mathbf{A}$ is positive semidefinite if and only if it is a covariance matrix. Then I need to verify in both directions, i.e. ...
1
vote
1answer
48 views

Conditions under which a known vector valued function the gradient of some function

Suppose that we have a vector valued function $D(x)$ with derivative $H(x)$ and that both of these are smooth. Under what conditions does there exist a function $f(x)$ such that $\nabla f(x) = D(x)$? ...
2
votes
2answers
50 views

Conditions when a permutation matrix is symmetric

I am now playing with permutation matrices, http://mathworld.wolfram.com/PermutationMatrix.html. Also, there is a similar discussion: Symmetric Permutation Matrix. I want to ask more details than ...
-1
votes
0answers
27 views

Dual of an Algebra

I want to know that, as I know if I have two algebras and the morphism between them is an algebra morphism ... i am trying to figure out dual of that algebra morphism. For this I just need to take ...
0
votes
0answers
54 views

question about linear programming minimization

(P) $\min z=x_1+x_2$ subject to : $ x_1+2x_2 \geq 4$ ( equation 1) $2x_1+x_2\geq6$ (equation 2) $-x_1+x_2\leq1$ (equation 3) $x_1>=0 ,x_2\geq0 $$ $ I'm trying to solve this using two-...