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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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35 views

Eigenvalues and eigenvectors of reflection matrix

I'm working on the following problem: Let A be a reflection matrix, such that, $a_{ij}=\delta_{ij}-2n_{i}n_{j}$, about a plane perpendicular to $\vec{n}$, $\vec{n}$ being the unitary vector. Find ...
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2answers
24 views

Show: $\sum_{i,j=1}^n |a_{ij}|^2=trA^*A=\sum^n_{i=1}\sigma_i^2$

I'm trying to prove the above fact for an arbitrary matrix $A$, with eigenvalue $\lambda_i$, and singular values $\sigma_i$. My approach so far: the trace of a matrix is the sum of its eigenvalues, ...
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2answers
22 views

Is it possible to apply cofactor expansion to a $2$x$2$ matrix? Why is the determinant of a constant seemingly one?

Solved: the determinant of a constant has the same value as that constant. It is possible to apply cofactor expansion to a two-by-two matrix. For example, this matrix: $$ \left[ \begin{array}{cc} ...
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0answers
9 views

Subspace span from Reduce Row-Echelon

Let $A$ be a matrix in $\mathbb R ^ {5 \times 7}$ of rank 3, so the spans of the columns and rows of $A$ are $3$-dimensional subspaces of $\mathbb R^5$ and $\mathbb R^7$, respectively. My problem is: ...
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0answers
44 views

Number of unique paths on the edges of a grid with wraparound that return to the origin

I was given this problem on the codegolf stackexchange, but I don't know where to begin on how to calculate it, except by creating some brute-force program to do it for me (like almost all existing ...
2
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2answers
58 views

Determinant and trace of a matrix

Find $\det A$ and $\text{Tr} A$ for the matrix $A\in M_n(\mathbb{Q})$ such that $\sqrt[n]{p}$ is an eigenvalue of $A$, where $p$ is a prime number or a positive integer such that the square root is ...
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2answers
32 views

Does polynomial keep inverse?

Let $A=(a_{i,j})_{n\times n}$ be an invertible matrix with the positive rational entry. Let $p(x)$ be a rational polynomial. Consider the following matrix \begin{align*} B=\left(p(a_{i,j})\right)_{n\...
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2answers
18 views

A line with a parametric vector equation

I am struggling with this question. Any help would be greatly appreciated
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1answer
23 views

Find displacement vector AB, given A is point (-3,5,7) and B is (-5,-4,9)

I am unsure if I answered this questions correctly. If A is a point with coordinates (-3,5,7) and point B has coordinates (-5,-4,9) then find displacement vector AB. I got (-8,1,16), is this correct ...
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4answers
85 views

$A^2 + A + I_n = 0 \implies$ matrix $A$ is invertible

I need to proof that if $A^2 + A + I_n = 0$ then matrix $A$ is invertible. I can see why $A^2 + A$ is invertible, but can't find a way to proof it on $A$.
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0answers
17 views

How to “combine columns” in a linear system with dependent unknowns?

I'm having some trouble to understand what combining columns means in this text: The block-diagonal system in ($3.194$) was defined region-wise without regard to the boundary conditions on the ...
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1answer
16 views

Linear maps and matrix representations

I have a question. I know that every linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be expressed as $T(x)=Ax$. Where A is a matrix. Does this hold for any basis or just the standard basis? I ...
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1answer
51 views

How to calculate correctly the gradient of log det $\psi(x)$ matrix?

I have a function: $$ \psi(t) = \frac{1}{2} \ln det\left(-\frac{\Omega(t)^{-1}}{2}\right) $$ where $\Omega(t)$ is a symmetric matrix semi-definite positive of size $k$. When I want to calculate the ...
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2answers
57 views

How to generate a rotation matrix given an angular momentum matrix

In 3 dimensions, the total angular momentum (for $z$) matrix is given. It generates the rotation matrix around $z$ by $e^{-i\theta J_3/h}.$ My question is how do we actually go about doing this? I ...
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1answer
22 views

Proof and notations

I was reading this proof, and I do not know what does E stands for, could you help me please? Theorem: Raising an adjacency matrix A of simple graph G to the n-th power gives the number of n-length ...
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0answers
14 views

Developing and Expression for Game Probability and Initial state vector

The question is: "Suppose you start the game with 20 coins, and your friend starts with 30. What is the initial state vector? " Since I have 50 coins, I know it is possible to have anywhere between ...
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1answer
44 views

Interpretation of Einstein notation for matrix multiplication

Consider the matrix product $C = AB$ where $A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{n \times p}$. The Einstein summation notation for this is $$ c_{ik} = a_{ij}b_{jk}. $$ Is there any ...
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0answers
13 views

Solving a linear equation using columns and rows of matrix

I have a system of equations that I am trying to approx. an answer to. I don't do any high level mathematics, so excuse any formatting errors. Find every team's individual match score. vars = $...
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1answer
33 views

Simpler terminology for determinant and other matrix terms [closed]

Wondering what words you could use to describe the following Matrix properties. Determinant Eigenvalues and Eigenvectors Identity Trace Adjugate Cofactor Conjugate transpose The prefix eigen- is ...
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3answers
52 views

Prove that $\det(A A^t) =0$ for every $4 \times 3$ matrix $A$

I try to prove that the determinant of $A A^t$ equals $0$. $A^t=(A \text{ transpose})$. I tried to prove it by contradiction. if $\det(A A^t )\neq 0$ so there exists an inverse, so I tried to ...
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1answer
26 views

Determining a point on a plane given a normal and a vector that passes through a point

I'm really stuck on determining the constraints of this question in part 2. Part one states: ...
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1answer
24 views

Finding the matrix of $T(p(x)) = p(2x-1)$ with respect to the basis $B = $ {$1+x, 1-x, x^2$}

Finding the matrix of $T(p(x)) = p(2x-1)$ with respect to the basis $B = $ {$1+x, 1-x, x^2$} To find the matix of a transformation with respect to a given basis, I find the images of the basis ...
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0answers
18 views

Finding minimum of sum of matrix elements

Consider a 'row' which can be in one of $n$ states. I have an $n \times n$ matrix $M$ such that $M(i,j)$ gives the interaction energy between a lower row in state $i$ and a row in state $j$ placed ...
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2answers
60 views

What am I doing wrong calculating natural logarithm of this matrix?

$$ A=\begin{bmatrix} 4 & -15 & 6 \\ 1 & -4 &2 \\ 1 & -5 &3 \end{bmatrix} $$ Find natural logarithm of $A$. I notice $Sp_A=\{1,1,1\}$. So $(A-E)^3$ must be zero. $\...
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0answers
21 views

Eigenvalues of a special product

Let $M$ be a $3 x 3$ complex matrix that is diagobalizable with eigenvalues $0$ and the other two eigenvalues on the unit circle. Let $m_1, m_2, m_3$ be positive integers such that $gcd(m_1,m_2,m_3) =...
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4answers
71 views

Developing a general expression for powers of matrices

Is there a relationship for matrices that are squared? I am trying to determine two possible matrices $P$ where $$P^2=\begin{bmatrix}0.6&0.4\\0.4&0.6\end{bmatrix}$$ I know that $P$ has to be ...
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2answers
262 views

Do similar matrices have same characteristic equations? [closed]

Since similar matrices have same eigenvalues and characteristic polynomials, then they must have the same characteristic equation, right?
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0answers
23 views

Show that any zero trace matrix is ​similar to a null diagonal matrix.

Show that any zero trace matrix is ​​similar to a null diagonal matrix. let $A=(a_{i,j})_{1 \leq i,j \leq n} \in M_n(K)$ such that $ \sum_{k=1}^{n} a_{k,k}=0$ I need to show that that there is $P \...
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1answer
22 views

Linear transformation with range of a vector [duplicate]

enter image description here What are the steps or any guidance in solving this problem? I'm very confused
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1answer
25 views

Definite positive matrix

According to a book that I have an hermitian matrix is definite positive if $X^TA\overline{X}>0$, but here in the forums (and other sources) the definition is given by $X^*AX>0$. Somehow I'm ...
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1answer
38 views

Invertible matrices of type $A+\varepsilon I$

It is well known that $\det$ function is continuous and, by $$ \det.^{-1}(\mathbb{C}^{*})=GL_n(\mathbb{C}),$$ the set $GL_n(\mathbb{C})$ is a open subset of $M_n(\mathbb{C})$. From here, some authors, ...
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1answer
25 views

Constructing matrices that are diagonalizable or not

What are some methods where I can construct a matrix that is diagonalizable or not quickly for some constraints such as it being singular or not or it having complex eigenvalues for 2x2 and 3x3 matrix....
3
votes
2answers
71 views

Show a specially defined matrix is positive definite

Let $E_1, ..., E_n$ be non empty finite sets. Show that the matrix $A = (A_{ij})_{1 \leq i, j \leq n}$ defined by $A_{ij} = \dfrac{|E_i \cap E_j|}{|E_i \cup E_j|}$, is positive semi-definite. This is ...
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1answer
26 views

For what value of $k$ does system of Equations have infinitely many solutions? Gaussian Elimination - is my approach correct?

$$ x + ky + z = 1$$ $$ - y + z = 2$$ $$x + y + 2z = 3$$ Using Gaussian elimination I reduced the augmented matrix to: [\begin{bmatrix} 1&1&2&3\\ 0&-1&1&2\\ 0&0&k-2&...
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0answers
26 views

Determinant of a symmetric Hankel matrix with non-zero diagonal elements [on hold]

If $S_r= a^r + b^r + c^r,$ then what is the determinant of the following $3\times 3$ matrix? \begin{bmatrix} S_0 & S_1 & S_2\\ S_1 & S_2 & S_3\\ S_2 & S_3 & S_4 \end{bmatrix}
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0answers
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If $A$ and $B$ are transition matrices such that $||A-B|| < c$, then what can we say about $||A^n-B^n||$ for a given $n$?

Suppose there are two matrices, $A$ and $B$, that are both transition matrices for a Markov chain ($n\times n$, non-negative and row-stochastic). I know that A and B are "close" in the sense that $||A-...
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1answer
23 views

Why can we replace dependent variable y with the residuals/error term e?

I don't understand why we can replace $y$ with $e$: As in the proof of the Gauss-Markov theorem, $$ \tilde{\beta} = [\,(W' W)^{-1}W' + C\,][\, W\beta + e\,] = \beta + (W' W)^{-1}W'e + CW\beta + ...
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2answers
33 views

Symmetrical matrix determinant

Is it true that every symmetrical matrix has a determinant non-zero? If so how can I prove it? Note: A symmetrical matrix that is not zero. (thanks to the commenter that pointed it out)
3
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2answers
25 views

Is it possible for an eigenspace to have more than one vector in its basis? What would that imply? Would every vector in the basis be an eigenvector?

So far I've been seeing that the vector that makes up a basis for the null space of the matrix $A-Ix$ (where x is an eigenvalue) is the eigenvector corresponding to the eigenvalue $x$. But I've never ...
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1answer
54 views

Is there any matrix $A$ for which $A^{\dagger}A=AA^{\dagger}$ but $A\neq A^{\dagger}$?

I encountered the definition of a normal matrix: $$A^*A = AA^*$$ Whereas the definition of a hermitian matrix is: $$A^{\dagger}=A$$ From this it follows that $A^{\dagger}A=AA^{\dagger}$, very ...
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1answer
29 views

Creating a connected undirected graph with connectivity k

I am working on a computing mini-project and I am stuck because of the following problem: "Given a number of nodes N and a connectivity k such that $N > k + 1$, is it always possible to construct ...
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0answers
26 views

What does it mean when a matrix has functions as elements and not scalars?

As far as I understand, the matrix $$ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} $$ refers to a linear transformation of the ...
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1answer
20 views

Expression for $\|Ax\|_2^2$ in terms of rows of $A$

I am currently reading 1011.3027, Roman Vershynin's introduction to non-asymptotic random matrices. On page 24, there is an equation, (5.24), which reads \begin{align*} \|Ax\|_2^2 = \sum_{i=1}^n\...
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0answers
11 views

Unclear proof in the Gauss-Markov theorem

I don't understand a step in the proof of the Gauss-Markov theorem: Mainly, why can we simply replace y with e, given that y is defined as: Thanks in advance!
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2answers
29 views

Sum of symmetric, positive semidefinite matrices

Let $A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{p \times n}$. Show that $A^{T}A+ B^{T}B$ is invertible if and only if $\ker A \cap \ker B =\lbrace 0 \rbrace$. I could show that if it's ...
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0answers
12 views

Uniqueness of Degenerate Eigenfunctions

Say you have some matrix and the eigenvalues are all the same (degenerate). Let’s say you have $N$ eigenvectors with the label $n$, call them $\psi_n$. If the eigenvalues were not degenerate, then the ...
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1answer
43 views

Matrix Differentiation (involving Hadamard products)

I am trying to differentiate over the following Frobenius Norm: $$\Phi =||A-(B\circ C)D ||^2_F$$ with respect to B, C, D respectively, i.e.: $$\frac{\partial \Phi}{\partial B}, \frac{\partial \Phi}{\...
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3answers
126 views

How to solve $A^{\frac 12} B A^{\frac 12} = C$ for $A$?

Suppose that matrices $A,B,C$ are symmetric and positive definite. Then, $A$ has a unique, positive square root, which we call $A^{\frac 12}$. If $$A^{\frac 12} B A^{\frac 12} = C$$ then can we write ...
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1answer
32 views

Minimization of norm distance using SDPs, cone programming, etc.

Building on top of this question Minimization of Frobenius norm using SDPs, I would like to know for what class of norms ($ \Vert \cdot \Vert $ in the following) can we represent the following problem ...
3
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2answers
59 views

Is it true that $A^TA=A \implies A^2=A$? Is the converse true?

I was asked this question in an exam. Let $A$ be a square matrix. $A^TA=A \implies A^2=A$, true or false? $A^2=A \implies A^TA=A$, true or false? I rewrote the equations as $(A^T-I)A=0$ and $(A-I)A=...