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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2
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0answers
25 views

Find solution to matrix sandwich product [duplicate]

For any two $n \times n$ real symmetric and positive definite matrices $B$ and $C$, is it always possible to find a third real symmetric and positive definite matrix $A$ such that $ABA=C$? If not, ...
0
votes
0answers
11 views

Complex Matrix Orientation

I recently learned about the fact that a linear mapping of a real vector space is orientation-preserving if the determinant of the matrix is positive. Now I was wondering if there exists a similar ...
0
votes
0answers
16 views

Squaring Orthogonal Matrix Entrywise is Doubly Stochastic

I'm stuck on this question: Suppose Q is an orthogonal matrix. Show that $Q \circ Q$ is doubly stochastic, i.e. entries are nonnegative, every row sums up to 1, and every column sums up to 1. (the ...
1
vote
2answers
22 views

Determine the values of $x$ for which$ M^{-1}$ does not exist

$$ \text{Let} \qquad M=\begin{bmatrix} x(x^2-1)&x \\ 3&1\end{bmatrix} \quad.$$ I have to determine the values of $x$ for which $M^{-1}$ does not exist. When $x=0$ and $x=2$ the determinant ...
0
votes
1answer
22 views

Scalar derivative of quadratic form where matrix depends on variable

I have the expression $$K(p(t),q(t)) = p^T D(q) p$$ Where D(q) is an n x n symmetric matrix, q and p are vectors (n x 1) depending on scalar variable t. I need to take the derivative of K with ...
0
votes
1answer
33 views

Numerically Integrate Matrix Equation in Matlab / Octave

How do I integrate a matrix numerically in Octave / Matlab? I am trying to do the following integration numerically in Matlab $\int_a^b T(x)^{-1} B dx$ where $T(x)$ returns an 8x8 matrix. The ...
0
votes
3answers
38 views

Positive semi-definite matrix has a non-negative trace?

A simple question: If $A$ is a positive semi-definite matrix ($A\succeq 0)$, does it imply $Tr(A)\geq 0$, where the $Tr(\cdot)$ denotes the trace. If not, any counter-example? Thanks.
1
vote
1answer
30 views

Generating the special linear group of 2 by 2 matrices over the integers.

Our Number Theory professor claimed that the special linear group $\text{SL}_2(\mathbb{Z})$ is generated by just two matrices: $$ M_1=\begin{pmatrix} 0& -1\\ 1& 0 \\\end{pmatrix} $$ $$ ...
1
vote
1answer
11 views

Could the multiplication of matrix X (with dimensions [d+1 x N]) and its transpose simplify to a matrix with [d+1 x d+1] dimensions?

In a machine learning course I'm taking, one of the lectures deals with matrix multiplication. Could anyone explain why the dot product of these two matrices would "shrink" to [d+1 x d+1] ...
3
votes
1answer
52 views

If $CD = -DC$ Show that either $C$ or $D$ has no inverse.

I'm probably missing something obvious, but how would I go about solving this?
0
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0answers
13 views

Common or distinct covariance matrices for mahalanobis distance classifier

I'm very new to pattern recognition so apologies in advance for a newbies question. I've been experimenting with mahalanobis distance to class mean based classifiers to predict diagnoses from ...
1
vote
0answers
12 views

Is this matrix associated with an arbitrary group of events positive semi-definite?

Now I have an arbitrary group of events $X_1,X_2,\ldots,X_m$(with no independence or correlation assumptions, nor distribution knowledge), and define a symmetric matrix $\mathbf{K}$ as below: $$ ...
3
votes
2answers
85 views

What are the conditions should be added so that submatrix has full rank

Suppose a $6 \times 4$ matrix satisfies the following where $\alpha, \beta, \gamma, \theta, \sigma, \mu$ are non-zero. What are the conditions should be added so that any $4 \times 4$ submatrix has ...
0
votes
1answer
12 views

rank of a matrix which is a concatenation of full rank matrices

Suppose a $6 \times 4$ matrix satisfies the following where $\alpha, \beta, \gamma$ are non-zero. Is it true that any $4 \times 4$ submatrix also has full rank? I think it is true as I can't find a ...
0
votes
0answers
9 views

Cayley-Hamilton type decomposition of SL(3,R) matrices

Given an element $\lambda = \theta_a T_a$ of SL(3,R) Lie algebra, where $T_a$s are the generators and $\theta_a$s are parameters, is there a general formula to determine the coefficients A,B and C ...
0
votes
1answer
25 views

How to get characteristic polynomial of adj$(A)$

Suppose $\mathop{A}\limits_{n\times n}$ be a singular matrix which characteristic polynomial is $\psi_A(x)$. How to find the characteristic polynomial for adj$(A)$ ? I know what to do for ...
1
vote
0answers
16 views

Calculating the left pseudoinverse of a Matrix whose columns are Probablity Mass Functions

I have a matrix $A_{m\times n}$, where $A_j$ , a column of $A$ represents a probability mass function, and so the sum over the column is 1. This is true for all the columns of A, i.e. $\forall j \in ...
0
votes
1answer
25 views

Determine the values of x to make this matrices not work in inverse

\begin{pmatrix} x(x^2-1) & x \\ 3x & 1 \end{pmatrix} This is a $2\times 2$ matrix $M$. Determine the values of $x$ for which $M^{-1}$ does not exist (Inverse) I figured that $0$ ...
1
vote
1answer
16 views

Figuring out the variables rows and colums for matrices

Let P be a 2 × 3 matrix, Q a m × 5 matrix and R a p × q matrix. Find the values of m, p and q such that the operation Q - PR is possible. So I figured that p = 3 Is m=2 and q=5? Just need to make ...
0
votes
0answers
28 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
vote
1answer
78 views

Can matrix exponentials ever be negative? If so, under what conditions?

Let $C$ be a $2 \times 2$ matrix with real entries, and $x\in\mathbb{R}^2$. We write $x > 0$ if both coordinates are strictly positive. Suppose $x>0$, under what conditions on $C$ and $x$ ...
1
vote
0answers
12 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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
0answers
32 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 ...
0
votes
0answers
23 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
votes
0answers
21 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 & & ...
0
votes
2answers
27 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 ...
2
votes
2answers
24 views

Forming a new matrix by adding the same number to any row or column

Say that two $m\times n$ matrices, where $m,n\ge 2$, are related if one can be obtained from the other after a finite number of steps, where at each step we add any real number to all elements of any ...
0
votes
0answers
29 views

Matrix that needs to be reduced to reduced row echelon form

What does the first column mean? Do I move the first column to the last column?
1
vote
1answer
21 views

rank of a submatrix

Suppose the $8 \times 4$ matrix $A$ has rank $4$. Is it always true that any $4 \times 4$ submatrix of $A$ has rank $4$? I am doing research on coding theory and I am wondering whether this is true. ...
-2
votes
0answers
29 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 ...
1
vote
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} ...
1
vote
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 ...
3
votes
2answers
50 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 ...
0
votes
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 ...
5
votes
3answers
258 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 ...
0
votes
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?
0
votes
1answer
22 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 ...
3
votes
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*} ...
4
votes
1answer
79 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 ...
0
votes
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 ...
-1
votes
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 ...
0
votes
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 ...
3
votes
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 $q$. ...
0
votes
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 ...
0
votes
1answer
26 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 ...
1
vote
1answer
17 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, ...
0
votes
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? ...