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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Analytic methods of computing entrywise absolute values of matrices

Are there any analytic ways of computing, or closely approximating the entry wise absolute value of a matrix? More explicitly, for an arbitrary matrix $B\in \mathbb{R}^{m\times n}$, are there any ...
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Is there any situation where the LDU decomposition is the same as the eigenvalue decomposition?

I was just wondering if there are any situation where the LDU decomposition is the same as eigenvalue decomposition (diagonalization)? The only way this can be possible if L and U are inverse so ...
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Understanding the proof of the condition number of a matrix.

I am trying to understand how the condition number of a matrix is deduced from a linear system of equation. I have a system of equations $A \vec{x} = \vec{b}$ to which we introduce certain ...
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What is the name of the matrix operation “$\;\overrightarrow{y}\;$”?

I was doing some stuff for my computer science class, and it was talking about the vectorized form of unregularized logistic regression and I'm pretty bad at matrices and when I saw this symbol I was ...
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Strong linear independent of matrices

Let $A_1,\dots, A_n$ be $n\times n$ (real) matrices. What is a necessary and sufficient condition so that there exists a row vector $r^t\in \mathbb{R}^n$ such that $$ r^tA_1,\dots,r^tA_n $$ are ...
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Variation of Hertzsprung's problem

Image of the problem How to represent this situation Mathematically?
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1answer
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Distributive property of matrix-vector multiplication?

I know that matrix multiplication is not distributive but what about matrix-vector multiplication? If $A \in \mathbb{R^{m \times n}}$ and $\vec{x}+\delta \vec{x} \in \mathbb{R^{n \times 1}}$. Then can ...
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How Does One Use a Rotation Matrix to Calculate the Rotation About an Arbitrary Axis?

I have a rotation matrix for a body rotating about some arbitrary axis. I can determine the axis of rotation and the corresponding magnitude of the rotation. But I would like to know how to ...
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$m$ students in $n$ clubs and every pair of students can be found in one and only one club

$m$ students form $n$ clubs. Each student can be in multiple clubs, however, every club must contain exactly $n$ students. If every pair of students can be found in one and only one club, what can be ...
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30 views

Linear independence of solutions to Floquet equation

I have a question about solutions to the Floquet equation, i.e.: $$ \dot{x} = A(t) x $$ where $x$ is an $N$-dimensional time-dependent column vector and $A(t)$ is a $T$-periodic $N \times N$ matrix. ...
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How can we find the determinant of $2A^TA$ while only knowing the determinant of A and the order of the matrix? [closed]

If $A$ is a square matrix of order 3 and $det(A)=5$, then how much is $det(2A^TA)$? Assuming the product of a matrix and its transpose is nothing special how do we solve this question? This was a ...
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How to calculate determinant? [duplicate]

How to calculate the following determinant: $\begin{vmatrix} a & b & b & ... & b \\ b & a & b & ... & b \\ . & . & . & . ... & .\\ b & b &...
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What does one mean by: “Minimize weighted angles for shoulder and wrist.”?

English isn't my primary language, so I had no idea what he was talking about. I am working on my own inverse kinematics solver. It solves for 2 limbs (arm), in 3D space. I want the it to move ...
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1answer
55 views

How to find the determinant of a matrix of sines and cosines?

Find the value of the determinant of the following matrix. $$D= \begin{pmatrix}\cos a & \sin a & \cos a & \sin a \\ \cos 2a & \sin 2a & 2\cos 2a & 2\sin 2a\\\cos 3a & \sin ...
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Matrix monotonicity $A \ge B$ then $B^{-1} \ge A^{-1}$?

I am wondering if it is true that if in the sense of matrices we have $A \ge B$ that in case both of them are invertible, we also have that $B^{-1} \ge A^{-1}$?
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system of linear equations question (inferring number of solutions and rank of matrices).

So I have encountered this question and I had a hard time analyzing it, and didn't reach anything, here's the question: Let $A\in \mathbb{R}^{m\times n}$ and $Ax=0$ has only one solution. and $B\in ...
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How to find the minimal polynomial?

Let $$T : \mathbb{R}^{n \times n} \to \mathbb{R}^{n \times n}, \qquad T(A) = A^{t}$$ I want to find its minimal polynomial, but in order to do that I need to find the matrix basis $[T]_{\tilde{e}}$ ...
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Singularity and Dimension of a Matrix

So I'm currently working on a question that has me seriously confused about some of my previous linear algebra knowledge. I'm looking at the graph of two linear functions (call them $f$ and $g$) that ...
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Curl of $u=(u_1,u_2)$

If I have $u=(u_1,u_2)$ is a vector of two components, then what is $\text{curl}\,u:=\nabla\times u$? where $\nabla=(\partial_x,\partial_y)$. I think it should give a scalar and not a vector.
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Finding eigenvectors of a $2 \times 2$ matrix

For the following matrix, \begin{bmatrix}1&h\\0&1\end{bmatrix} I was able to find the eigenvalues to be $1$ with multiplicity $2$. How to find the eigenvectors?
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Product of matrices and vanishing trace

It is well known that if $A$ and $B$ are respectively symmetric and skew-symmetric, then $\text{Tr}(AB) = 0$. But is there some kind of analogous result for if one of these is hermitian? What I mean ...
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What is the computational complexity of solving a highly underdetermined systems

Let $F$ be a finite field with $q$ elements. Consider an underdetermined system of linear equations with $m$ equations and $n$ variables where $n>>m$. What is the complexity of solving such a ...
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28 views

Matrix from a linear application

I'm trying to prove these two important propositions from my monography. T is a linear transformation and $\alpha$ and $\beta$ are basis. I want to prove these two equalities Sorry for my poor ...
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$A$ is $p\times n$, $B$ is $q\times n$, what can we say about Systems $1$ and $2$ of equations?

Problem: Let $A$ be a $p\times n$ matrix, and $B$ be a $q\times n$ matrix. System 1: $Ax < 0, Bx= 0$ for some $x\in\mathbb R^n$. System 2: $A^Tu + B^Ty = 0$ for some non-zero $(u,y)$ with $u\ge 0$. ...
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44 views

Complex vector/matrix multiplication

I am currently studying a textbook and I have a limited background in complex linear algebra and unfortunately the textbook does not provide background knowledge for this either. As a result I am a ...
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1answer
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Why in scalar product I can multiply two vectors $\overrightarrow{a}_{1x3} * \overrightarrow{b}_{1x3} $ , whereas in matrices products I couldn't?

If a vector is a matrix of one dimention, Why in scalar product I can multiply two vectors $\overrightarrow{a}_{1x3} * \overrightarrow{b}_{1x3} $ , whereas in matrices products I'd need $a_{1x3} * b_{...
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Given $2 \times 2$ matrix A over field $F_p$. How many matrices of fixed determinant there exist?

Note: $p$ is prime number The idea that I had in mind was to fix the determinant $\Delta$. We can easily say what is determinant of matrix $2 \times 2$. If we put $A = \begin{pmatrix}a & b \\ c &...
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A strange characterization of the minimum singular value of a matrix

Let $A = \begin{bmatrix}a_1 | \cdots|a_n \end{bmatrix} \in\mathbb{R}^{n\times n}$. Is it true that $$\sigma_n(A) = \min_{1\leq j\leq n} \min_{\alpha\in\mathbb{R}^{n}}\|a_j - \sum_{i=1,i\neq j}^{n}\...
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$\left( {\begin{array}{cccc|c} 1 & 0 & 1 & 0 & 0\\ 1 & 1 & 0 & 0 & 1\\ 0 & 1 & 1 & 0 & 1\\ \end{array} } \right) $ in $\mathbb{Z}_{2}$

\begin{align} &\left( {\begin{array}{cccc|c} 1 & 0 & 1 & 0 & 0\\ 1 & 1 & 0 & 0 & 1\\ 0 & 1 & 1 & 0 & 1\\ \end{array} } \right) \text{in}\ \mathbb{Z}...
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32 views

Combination of two sets of basis

Given two sets of basis $V = \{v_1,...,v_n\}$ and $U = \{u_1,...,u_n\}$, each spanning $\mathbb{R}^n$, is it possible to select a subset of the two sets $T$ such that T = $\{v_1\, ...,v_m\} \cup \{u_1\...
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Equality of traces of two matrices

I was going through Boyd's EE363 HW 4 . In page 11,I found this I dont understand why the 2 expressions involving trace (in the MSE equation ) are equal to each other . $\Sigma_{x}$ is postive ...
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1answer
25 views

Linear map under certain conditions

Let $u_1 = (1,-1,1)$, $u_2 = (1,0,-1)$, $u_3 = (0,-1,1)$ and $u_4 = (2,-2,1)$. I want to determine all of the possible values for $a \in \mathbb{R}$ for which there exists a linear map $f:E \to E$ ...
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Matrix equation $A\cdot x=b$ where exists only one solution and $\det(A)=0$

The system of equations in matrix notation is \begin{align*} \underbrace{\begin{pmatrix} a_{11}-1 & a_{21}& a_{31} \\ a_{12} & a_{22}-1 & a_{32}\\ 1 & 1 & 1 \\ \end{pmatrix}}_{...
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Number of $2 \times 4$ grids whose rows and columns have even number of evens

Given a $2 \times 4$ grid, I need to find in how many distinct ways one can fill the grid with $1,2,\dots,8$ such that on each row and column there are an even number of even numbers. Repetitions are ...
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Find the smallest eigenvalue of a matrix via the power method

Find the smallest eigenvalue of the following matrix via the power method. $$\begin{bmatrix} 2 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 2 \end{bmatrix}$$ Accurate up to $3$ ...
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1answer
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Does a matrix A and its reduced row echelon form R have the same nullspace?

I was thinking in terms of this. Consider the general solution to $Ax = 0$, represented in terms of free variables. Setting all but one of the free variables to 0, we get a vector in the nullspace of ...
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If $x^{T}A^{T}Ax = x^Tx$ holds for every $x$, then $A^{T} A = I_n$

Given $A \in \mathbb R^{n \times n}$, if $$\left( \forall x \in\mathbb R^n \right) \left(x^{T} A^{T} A x = x^T x \right)$$ how to conclude that $A^{T}A = I_n$? I appreciate any help!
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Is the null space of a matrix the same after you multiply it by more matrices?

Suppose I know that the nullspace of matrix $P$ is $\mathbf{c}$. Now suppose I multiple $P$ by some matrix $C$, ie. $CP$. Is the nullspace of $CP$ also just $\mathbf{c}$? Or can there be other vectors ...
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Under what conditions of $A,B$ we have $\det (A + \lambda B)$ is uniformly $0$ for all $\lambda$?

Assume $A,B$ are real-value symmetric matrix (maybe redundant condition) and $\lambda \in \mathbb{R}$. This question comes from here, but the answerer doesn't give the condition.
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Solving system of polynomial matrix equations over $\Bbb Z_2$

Let $A, B, C, D$ be $4 \times 4$ matrices over $\mathbb{Z}_2$. Suppose they satisfy \begin{equation} \begin{cases} A^2+ BC+ BCA+ ABC+ A= I_4\\ AB+ BCB+ ABD=0_{4 \times 4} \\ CA+ ...
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Matrix Operations with $(A+BC)^{-1}$

so I know the rule $$(AB)^{-1} = B^{-1}A^{-1}$$ but is there any way to manipulate $$(A+BC)^{-1}$$ I really am trying to get better at Matrix Operations so I am practicing below where all $n \times n$ ...
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1answer
53 views

Matrix with determinant 1

How do you show that any $2 \times 2$ matrix \begin{bmatrix} a & b\\ c & d \end{bmatrix} with $a,b,c,d$ non-negative integers such that $ad - bc = 1$ is generated by \begin{bmatrix} 1 & 1\\...
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21 views

For n odd, skew symmetrical matrices in $M(n,\mathbb{R})$ are homeomorphic to orthogonal matrices for which $\det(A+I) \neq 0$?

I have a continuous mapping $A \mapsto (A-I)(A+I)^{-1}$ between the set of orthogonal matrices $A^{t}A=I$ s.t. $\det(A+I) \neq 0$ and the set of skew symmetrical matrices $A^{t}=-A$ in $M(n, \mathbb{R}...
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1answer
51 views

Jordan form of the exponential of a matrix

Given a matrix $A$ which has the form of a Jordan normal matrix: $$ A = \left(\begin{array}{cccc} \log(\phi_1) & 1& ...& 0\\ & \log( \phi_2)& & \\ & & \ddots &...
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Find the state vector of the regular Markov chain whose transition matrix is given by [closed]

https://www.chegg.com/homework-help/questions-and-answers/find-steady-state-vector-regular-markov-chain-whose-transition-matrix-given-75-25-t-answer-q28950470?trackid=544c4ecf7bd8&strackid=...
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Questions about Moore–Penrose inverse

I have some questions about Moore–Penrose inverse. Let $A, P\in \mathbb{R}^{d\times d}$. Suppose $A$ is positive definite and $P$ is a projection matrix with $P^2=P, P^\top=P$. I try to prove that $A^...
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1answer
51 views

Show that exponential map between complex matrix and general linear group is surjective

$\textbf{Problem}$ Trying to that $\exp: M_n(\mathbb{C}) \rightarrow GL(n,\mathbb{C})$ is surjective. $\textbf{Attempt}$ I started by considering a complex matrix A $\in M_n \mathbb{C}$, to this ...
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11 views

Time complexity of Kronecker product $I_n \otimes Q$ where $Q$ is Toeplitz

Given a symmetric Toeplitz matrix $Q$ of size $m$, what is the time complexity of the Kronecker product $I_n \otimes Q$, where $I_n$ is identity matrix of size $n$? Is is $\mathcal{O}(m)$?
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25 views

How to relate Frobenius norm and trace of a matrix?

I came across an expression relating the Frobenius Norm with Trace as follows : $$\|UU^T\hat X\|^2_F = tr((UU^T\hat X)^T(UU^T\hat X))$$ Could someone explain briefly the equivalence ? THanks

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