For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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33 views

algorithm game theory

$$\mathbf{A}=\mathbf{B}\mathbf{T}= \begin{pmatrix} 3& 3 & 0 \\ 4 & 0 & 1 \\ 0 & 4 & 5 \\ \end{pmatrix} $$ Every Nash equilibria of this ...
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0answers
27 views

Positive Definite Matrices Properties that I'm trying to prove right/wrong:

Is the product of two: (a) positive definite matrices positive definite? (b) symmetric positive definite matrices positive definite? (c) symmetric positive definite matrices symmetric positive ...
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0answers
27 views

Positive Definite Matrices, HW. [duplicate]

Is the product of two: (a) positive definite matrices positive definite? (b) symmetric positive definite matrices positive definite? (c) symmetric positive definite matrices symmetric positive ...
1
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0answers
11 views

A question regarding the proof of Laplace's expansion on Wikipedia

I am reading the proof of Laplace's expansion on Wikipedia and have a dilemma regarding the following: $\tau = (n, n-1, \ldots, i) \;\sigma^\prime\; (j, j+1, \ldots, n)$ As far as I know, such ...
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0answers
31 views

Solving system of equations over $\mathbb Z_{3}$

I need to solve this system of equations over $\mathbb Z_{3}$ using matrix row reduction and write it in parametric form. I have an answer but I've been having trouble so I am really looking for ...
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0answers
60 views

Reducing a linear algebra expression to quadratic form

I am trying to solve the following exercise for my Machine Learning course. Expand this expression so that there are only quadratic terms: $(\mathbf{x} - \mathbf{\mu})^T \mathbf{\Sigma}^{-1} ...
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0answers
33 views

Integration of ODE equation in Matlab / Octave

I have a system of 8 ODE's where the initial conditions are in matrix form. $\frac{dT}{dS} = H T$ where T at the initial state is the identity matrix. $T(a) = I$ H is a constant 8x8 matrix T is ...
2
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0answers
758 views

SVD by QR and Choleski decomposition - What is going on?

Here's an algorithm I found that performs Singular Value Decomposition. I preferred this algorithm because it can be parallelized, and I don't have to calculate the huge $AA^T$ matrix when the number ...
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1answer
21 views

How to construct matrix from 4 sub matrices in Matlab?

I have an 8x8 matrix defined as $T = \begin{bmatrix}T_{UU} \quad T_{UF}\\T_{FU} \quad T_{FF}\end{bmatrix}$ I can define $T_{UU}$, $T_{UF}$, $T_{FU}$, and $T_{FF}$ as ...
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1answer
38 views

Rotation Matrix to Quaternion(proper Orientation)

Given Data in the figure In this figure we have a unit vectors $ x,y,z$ as axis. Axis of rotation is $b$ and angle of rotation is $\phi$. $\phi$ is unknown and $b$ is given as $b= \frac{1}{2 ...
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0answers
30 views

Proof of uniqueness of reduced row echelon form

I've found a proof of uniqueness of reduced row echelon form. I have certian doubts with regard to this sentence: "It follows that R' and S' are (row) equivalent since deletion of columns does not ...
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22 views

Matrix, Gauss-Jordan Method

I have a application problem for math and I am unable to get all my system of equations. I have two of three. Celia had one hour to spend at the athletic club, where she will jog, play handball, and ...
0
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0answers
20 views

prove or disprove that a particular invertible matrix is also orthogonal

is it true that, if for some $2n \times 2n$ matrices $O^t=O^{-1}$ and $$J_0= \begin{bmatrix} \begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & ...
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2answers
25 views

Finding Cases Of Inequality Between Null Space And Solution Set

$H=(h\in R^m; Ah=0)$ $L=(l \in R^m; Al=b)$ Find a matrix $A^{n*m}$ so: $|L|=0 < |H|=1$ $|L|=0 < |H|=\infty$ $|L|=0 < |H|=7$ As for 1. \begin{pmatrix} 1 & 2 \\ 0 & 0 ...
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1answer
263 views

relation between size of matrix and condition number

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
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0answers
29 views
1
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1answer
744 views

Simultaneous diagonalization of two positive semi-definite matrices

Let matrices $A, B$ be two $n \times n $ positive semi-definite matrices and they can be represent in the following form $$A=\sum_{i=1}^{n} \psi_{i}p_{i}p_{i}^{T}=P\Psi P^{T}, \quad B=\sum_{i=1}^{n} ...
0
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1answer
277 views

gradient of vector 2-norm

I have a function $f(\Theta) = \frac{1}{2N}\| y-\mathcal{X}(\Theta)\|_2^2$. Matrix $\Theta\in\mathbb{R}^{m_1\times m_2}$, $y=[y_1,\cdots,y_N]^T\in\mathbb{R}^N$ is the observation vector, and we use ...
3
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2answers
128 views

Cross products?

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
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1answer
18 views

When are two 3D Lines parallel in Plücker matrix form?

When are two lines in 3 dimensional space parallel, when the lines are both represented by Plücker matrices $L$ and $L'$. I'm trying to prove the solution to this question: ...
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2answers
48 views

Singular Value Decomposition-noisy data

I have a system of the form $$Ay=f,$$ where $A$ is a $N\times4$ matrix, $y$ a 4-element array of unknows and $f$ an $N$-element array. I add Gaussian noise in my data. I tested the following ...
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0answers
73 views

What does a '$-$' mean in front of a matrix?

I feel ridiculous for asking this, but I can't seem to find a clear answer. Let $$U = \begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$$ Show that $U$, $-U$, $-I$ where $I$ is the $2 \times 2$ ...
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5answers
50 views

Matrices to the power of $n$ and their reversibility

Please forgive my ignorance. I am busy with a first year course in elementary linear algebra and there are some concepts I do not grasp. Particularly, questions regarding matrix invertibility. For ...
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2answers
48 views

example of complex structure with negative determinant

is it possible to find a matrix $J_1 \in GL(4,\mathbb R)$ such that $\det J_1=-1 $ and $J_1^2=-\operatorname{id}$ ? if it is, how can we prove that every matrix $M \in GL(4,\mathbb R)$ such that ...
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0answers
14 views

Simplify recursion function based on a matrix, real-world usecase

I have an auction running, and I'm trying to calculate the expected amount of first, second etc. places to be taken by a particular bid. To achieve that, based on historical data I make a following ...
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1answer
77 views

Why are matrices written as such?

Another thread has talked about the purpose of a matrix. Dr. Math roughly summarized it as: A matrix is just a compact notation, which allows you to specify several linear equations at once ...
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3answers
257 views

How to encode matrices uniquely

Given a square matrix $A=[a_{ij}]_{n \times n}$, an operation $swap(A, i, j)$ is defined to swap row $i$ and $j$ of $A$ and do the same thing with the corresponding columns. For example, in the ...
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0answers
50 views

Decomposing a stochastic matrix into a product of stochastic matrices.

It is well-known that any square real matrix of small rank $k$ can be decomposed into a product of a skinny matrix with $k$ columns and a fat matrix with $k$ rows by means of an SVD. This question is ...
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1answer
21 views

Transformation of a surface normal

I'm taking a university level course in discrete geometrics and graphical programming, and I'm having trouble understanding this exercise. Let p be a point in R^3, n a surface normal, and M a ...
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2answers
48 views

Flipping a matrix?

Real quick question: I was wondering, how would one denote mathemathically the flipping of a matrix, horizontally or vertically, around its own axis?
3
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1answer
35 views

What should be the characteristic polynomial for $A^{-1}$ and adj$A$ if the characteristic polynomial of $A$ be given?

Let the characteristic polynomial of $A$ be $\psi_A(x):=p(x)$. If $A$ be non-singular, then find that the characteristic polynomial of $A^{-1}$ and adj$(A)$. My attempt: We have \begin{align*} ...
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2answers
18k views

Transpose of inverse vs inverse of transpose

I can't seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix? Thanks!
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1answer
32 views

Solve $3$ variables using $4$ equations where $1$ equation contains $3$ variables

Suppose we are given the system of equations $$\alpha_1A+\beta_1B+\gamma_1C=x$$ $$\alpha_2A+\beta_2B+\gamma_2C+\theta_2D=y$$ $$\alpha_3A+\beta_3B+\gamma_3C+\theta_3D=z$$ where ...
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1answer
23 views

properties of largest eignvalue of product of two matrices

I'm searching for the proof of this lemma it's about largest eignvalue of product of two matrices. one of them is positive definete and the other one is symmetric. B is symmetric matrix, A is Positive ...
4
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1answer
103 views

Saddle Points on Matrices

Let $n$, $m$ be positive integers. Suppose that $A$ is a $2$ x $n$ or an $m$ x $2$ matrix and that it has a saddle point. Show that among the saddle points of $A$ there exists at least one which ...
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2answers
43 views

Is there a multiplication transformation that will add the bottom row of a matrix to the top row?

Given matrix $$ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} $$ Is there a matrix $B$ such that: $$ AB = \begin{bmatrix} a+g & b+h & c+i ...
4
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3answers
104 views

How to get determinant of $A$ in terms of tr$(A^k)$?

Suppose that $A$ is $n$-square matrix such that $t_r:=$ tr$(A^r), r=1, 2, \cdots, n$ are given real numbers. How shall we compute $\det(A)$ in terms of $t_r$s? I am completely unable to do this. ...
3
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1answer
13 views

Prove that $A$ is invertible when $a_0 \not=0 $ and $A^{-1}=q(A)$ for some polynomial $q$.

Let $p(\lambda)= (-\lambda)^n + a_{n-1}\lambda^{n-1} + ... + a_0$ be characteristic polynomial of matrix $A$. Prove that $A$ is invertible when $a_0 \not=0 $ and $A^{-1}=q(A)$ for some polynomial $q$. ...
13
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3answers
327 views

Theorem about positive matrices

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
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1answer
17 views

Infinite-dimensional version of Gram matrix is invertible

We all know that a Gram matrix (a matrix with entries that are inner products of basis functions) is a invertible. Suppose I have $a_{ij} = (h_i, h_j)_H$ where the $h_j$ are basis functions of a ...
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1answer
24 views

Cholesky Decomposition and Orthogonalization

I recently came across a methodology for orthogonalizing variables that are collinear, that uses Cholesky Decomposition, but I am not entirely grasping the intuition of it. Let' assume we have three ...
0
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1answer
18 views

What is the sample variance-covariance matrix?

This is a more succinct question from a previous post, but I have arrived at two different answers, and need help determining which - if either - is correct. I start with a 4*3 matrix: ...
0
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3answers
112 views

Find $P \in GL_3 (C) $ so that $^tP \cdot P* = A$.

Find $P \in GL_3 (C) $ so that $^tP \cdot P* = A$. $P*$ means the complex conjugated of $P$. A is a hermitian matrix: $ A = \begin{pmatrix} 1 & 0 &-i \\ 0&1&0 \\ i&0&2 ...
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1answer
16 views

Matrices admit a QR decomposition

I just wanted to ask which matrices admit a QR decomposition. I think that all matrices $A \in \mathbb{R}^{m \times n}$ with $m \ge n$ admit a QR decomp. Are these the only ones that have a QR decomp, ...
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1answer
3k views

Inverse of upper triangular matrix

I have an upper triangular matrix that I want to solve the inverse for. I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity ...
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1answer
39 views

What is the relationship between three points on a quadratic curve and the curves coefficients?

In other words, is there a formula to get the coefficients a,b and c in terms of three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3, y_3)$? I am asking this because I have a linear algebra problem that ...
0
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0answers
15 views

help me find the gimbal locks

I have this transformation (x, y, z) |-> (x'', y'', z''). How can the gimbal locks be discerned and where are they? ...
1
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1answer
25 views

How to find trace of adj$A$ from the characteristic polynomial of $A$?

Let the characteristic polynomial for $A$ be $t^n+c_1 t^{n-1}+c_2t^{n-2}+\cdots+c_{n-1}t+c_n$. From it, is it possible to find the trace of adj$(A)$ ?
8
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2answers
287 views

If $ \mathrm{Tr}(M^k) = \mathrm{Tr}(N^k)$ for all $1\leq k \leq n$ then how do we show the $M$ and $N$ have the same eigenvalues?

Let $M,N$ be $n \times n$ square matrices over an algebraically closed field with the properties that the trace of both matrices coincides along with all powers of the matrix. More specifically, ...
3
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1answer
74 views

a linear algebra problem arising in geometry

This is a matrix problem. Assume that $A$ and $B$ are real $n\times n$ matrices. Denote $\Lambda=A+iB$, $$ M=\left (\begin{array}{cc} A &-B\\ B & A \end{array} \right ) $$ I would like to ...