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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-3
votes
2answers
40 views

If $\det(A)=0$, must the null space of $A$ be zero? [closed]

Came along this question: If $\det(A)=0$ for an $N\times N$-dimensional matrix $A$, the null space of $A$ is equal to zero. True or false? Why? Thank you already!
5
votes
5answers
567 views

Why the determinant of a matrix with the sum of each row's elements equal 0 is 0?

I'm trying to understand the proof of a problem, but I'm stuck. In my book they consider that if all lines of a matrix has sum 0 then it's determinant is also 0. I checked some random examples and ...
-1
votes
1answer
13 views

Augmented matrix [on hold]

Animals in an experiment are to be kept under a strict diet. Each animal should receive $20$ grams of protein and $6$ grams of fat. The laboratory technician is able to purchase two food mixes: Mix ...
1
vote
1answer
60 views

markov chain computation

I consider a 2 state Markov chain: $X = \{1,2\}$, transitions are $M(i,j)$ and the matrix has a unique stationary distribution $\pi$: $$ \pi(1) = \frac{M(2,1)}{2-M(1,1) - M(2,2)} \\ \pi(2) ...
3
votes
3answers
294 views

Sylow $p$-subgroups of Finite Matrix Groups

Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$? I believe the answer is yes, because I ...
-2
votes
0answers
33 views

What is ${\sigma _{\varepsilon ,W}}(P)$? [on hold]

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
1
vote
2answers
36 views

Matrix Equation

Imagine the question: If $K$ and $L$ are $2\times 2$ matrices (knowing all of their components) and $KM=L$, solve for the matrix $M$. One simple solution is to set the components of $M$ as $x,y,z,w$ ...
15
votes
2answers
992 views

Why does the inverse of the Hilbert matrix have integer entries?

Let $A$ be the $n\times n$ matrix given by $ A_{ij}=\frac{1}{i + j - 1}$. I need to show that $A$ is invertible and the inverse has integer entries. I was able to show that $A$ is invertible. How do I ...
2
votes
2answers
65 views

Conditions for an orthogonal matrix equation

Let $B_1$ and $B_2$ be given $n \times n$ real non-singular matrices and consider the system of equations $$\begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\begin{bmatrix}B_1 ...
1
vote
0answers
38 views

How to solve the matrix equation $A\overrightarrow{x}=\overrightarrow{b}$ in Matlab when nullitity$(A)\neq 0$

Say, $A= \begin{pmatrix} 1 & 0 &1 \\ 0 & 1 &1 \\ 0 &0 &0 \end{pmatrix}$ and $\overrightarrow{b}= \begin{pmatrix} 8 \\ -5 \\ 0 \end{pmatrix}$ and I want to solve ...
6
votes
0answers
63 views

How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
0
votes
0answers
50 views

Finding permutation matrix

Let $P_{\pi}$ denote a permutation matrix associated to the permutation $\pi:\{1,...,n\}\rightarrow \{1,...,n\}$ and $\sigma$ denote the cyclic permutation $(1 2 ...n)$. If T is the $n\times n$ lower ...
3
votes
3answers
48 views

Find $p_{ij}^{(n)}$ for the transition matrix

Let $$P=\begin{bmatrix}\frac{1}{3}&0&\frac{2}{3}\\\frac{1}{3}&\frac{2}{3}&0\\\frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}$$ find ...
1
vote
1answer
24 views

Replacing pinv with inv in MATLAB

Let $\mathbf{y} = \mathbf{Ax}$ represent a system of equations where $A\in\mathbb{R}^{m\times m}, x\in\mathbb{R}^{m\times 1}$. However rank of $\mathbf{A}$ is $m-1$. I add another equation ...
1
vote
1answer
75 views

Compactness of a set of matrix polynomials with a norm restriction

Suppose $P_\Delta (\lambda) = (A_m + \Delta _m)\lambda^m + \cdots + (A_1 + \Delta_1)\lambda^1 + (A_0 + \Delta_0)$ is a matrix polynomial, and $\lambda $ is a complex variable. $A_j,\Delta_j \in ...
0
votes
1answer
20 views

Change from one cartesian co-ordinate system to another by translation and rotation.

There are two reasons for me to ask this question: I want to know if my understanding on this issue is correct. To clarify a doubt I have. I want to change the co-ordinate system of a set of ...
2
votes
1answer
38 views

Product of projections and commutativity

Let $P_1$, $P_2$, $\dots$, $P_m\in\mathbb{R}^{n\times n}$ be orthogonal projections projecting onto subspaces $V_1$, $V_2$, $\dots$, $V_m$, respectively, and let $P_{1\cap2\cap\dots\cap m}$ denote the ...
0
votes
1answer
33 views

What is $\max(\operatorname{Re} \{ \frac{x^* Ax}{x^* x}:0 \ne x \in C^n\} )$?

Let $A = \left( \begin{array}{*{20}{c}} 1&2\\ 0&1 \end{array} \right)$. What is $\max\left(\operatorname{Re} \left\{ \dfrac{x^* Ax}{x^* x}:0 \ne x \in C^n\right\} \right)$?
-1
votes
0answers
24 views

What is subordinate matrix norm?

What is 'subordinate matrix norm' in this question? .
0
votes
0answers
40 views

Wondering how to rotate a normal vector in 4 dimensions?

Saw another post that suggested a answer but need help with the answer and the other post is inactive. I know how to rotate in 3-space using matrix transforms for each axis no problem. Have a very ...
1
vote
3answers
70 views

Matrix Exponential and Logarithm

Consider the following matrix $A$: $A = \begin{bmatrix} \cos^2(1) & -\sin(2) & \sin^2(1) \\ \cos(1)\sin(1) & \cos(2) & -\cos(1)\sin(1) \\ \sin^2(1) & \sin(2) & ...
-1
votes
0answers
16 views

Affine Transform Matrix (Rotation) [closed]

Can someone help me with this question? I know how rotation matrix looks like after rotating by y axis but don't know how. Show how to derive the Rotation Matrix about the Y-Axis.
1
vote
1answer
31 views

Non square Matrix multiplication

Assuming we have the following matrix multiplication problem $$ {\bf A x} = {\bf b}$$ and that the dimensions of ${\bf A,x,b}$ are the following $3\times2$, $2\times 1$ and $3\times 1$ How can one ...
0
votes
4answers
41 views

What is the equation of a 3D line which represents the intersection between two 3D planes?

The intersection defined by the two planes $v \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$ and $v \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$ is a line. What is the equation ...
3
votes
1answer
29 views

A = UL factorization [duplicate]

How do I calculate $A=UL$ factorization where $U$ is upper triangular matrix with 1's along the diagonal and $L$ is lower triangular matrix? How is this similar to the $LU$ factorization?
1
vote
0answers
26 views

Cayley Hamilton Theorem using LU decompostion

I am trying to find the characterisitic equation of n*n matrix by Cayley Hamilton Theorem using LU Decompostion. Below is my algorithm to find U matrix. ...
2
votes
2answers
63 views

Calculate determinant of $ M=\left( \begin{array}{cc} A&-\vec d^T \\ \vec c& b \\ \end{array} \right) $.

I have a block matrix $$ M=\left( \begin{array}{cc} A&-\vec d^T \\ \vec c& b \\ \end{array} \right) $$ where $A$ is a $(n-1)\times (n-1)$ matrix, $\vec d,\ \vec c$ are two ...
2
votes
1answer
36 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
3
votes
1answer
28 views

Finding $[T_{|W_i}]_{C_i}$

Let $B=\{v_1,v_2,v_3\}$, a basis of $V$ above $\mathbb{R}$. Let $$ [T]_B = \left(\begin{array}{cccc} 6&-3&-2\\4&-1&-2\\10&-5&-3 \end{array}\right)$$ The characteristic ...
0
votes
1answer
32 views

Permutation and signature matrices “almost commute”

Let $\mathcal{P}$ be the set of all permutation matrices of order $n$ and $\mathcal{S}$ the set of all signature matrices of order $n$. Furthermore, let $$\mathcal{P}\mathcal{S} = \{PS \mid ...
-1
votes
0answers
43 views

What is Heine-Borel theorom

What is Heine-Borel theorom in this link?
5
votes
2answers
48 views

Why does the Hessian work?

I am working through Susskind's 'The Theoretical Minimum' (on physics) – it also includes some maths. In particular, there is an interlude for which he discusses partial differentiation. He discusses ...
0
votes
0answers
30 views

How to calculate $det(X^T X)$ efficiently, update one column of X each time

$X_{1} = (A, b)$, where $X_{1}$ is a $n\times p$ matrix, $A$ is a $n\times (p-1)$ and $b$ is $n\times1$. First calculate $\det(X_{1}^T X_{1})$, then update $b$ with $c$, st. $X_{2} = (A, c)$ and ...
3
votes
0answers
90 views

Spectral radius of a real, symmetric, positive semi - definite matrix.

While answering a question, the OP made a follow - up question, that I was not able to answer at that moment. However, I came up with an intriguing (at least to me) question. Let ...
2
votes
3answers
77 views

Diagonalize a symmetric matrix

let $$A = \left(\begin{array}{cccc} 1&2&3\\2&3&4\\3&4&5 \end{array}\right)$$ I need to find an invertible matrix $P$ such that $P^tAP$ is a diagonal matrix and it's main ...
1
vote
2answers
36 views

What am I doing wrong? - Change of basis matrix

Problem: Let $\alpha$ be the standard basis of $\mathbb{R}^3$ and let $\beta = \left\{(1,0,0), (1,1,0), (1,1,1)\right\}$ be another basis. Consider the linear map $T: \mathbb{R}^3 \rightarrow ...
18
votes
2answers
214 views

How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
0
votes
0answers
8 views

Generating Correlated Samples: Cholesky Decomposition of Correlation Matrix or Covariance Matrix? [duplicate]

I have multiple correlated stochastic processes and I would like to generate correlated samples of them. From my understanding, if I have my samples $Z$ and a Cholesky decomposition of their ...
0
votes
0answers
33 views

Help with textbook formula

In Bishop - Pattern Recognition and Machine Learning, Section 1, I do not fully understand Formula (1.65). Although it's not stated explicitly, I assume that I is the identity matrix with the ...
-6
votes
0answers
47 views
2
votes
2answers
46 views

Realizing the oscillator algebra as a matrix Lie algebra

In Hilgert's & Neeb's Structure and Geometry of Lie Groups, they introduce a Lie algebra, which they call the "oscillator algebra," as an extension of the Heisenberg algebra. They give a basis ...
1
vote
1answer
59 views

Proving Properties of Markov Chain

I want to prove that the queue length at a store is not a Markov Chain. $Q_k$ is the queue length at time instant $k$, $V_k$ is the number of arrivals. At every time instant one customer is ...
0
votes
1answer
47 views

Transition Probability matrix for on/off swtiches

Lets say we have $5$ gates. On state is designated by 1 and off state is designated by $0$. Then, $\{ W_k^i, k =0,1,2,\ldots\}$ is a discrete parameter Markov chains for each gate $i, \{ i = ...
0
votes
2answers
623 views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
2
votes
2answers
47 views

Is rank$(AQB)=$rank$(AB)$ if $Q$ is non-singular?

$\newcommand{\rank}{\operatorname{rank}}$We know that $\rank(PA)=\rank(AQ)=\rank(PAQ)=\rank(A)$ where $A\in M_{m\times n}(\mathbb F), P, Q$ are $m\times m, n\times n$ invertible matrices. mean to ...
-1
votes
1answer
47 views

Column vector of simultaneous equaations' solution

Struggling with some basics of Linear Algebra. Please help. Let's restrict the discussion to 2D space & consider the following simultaneous equations: $2x + 3y = 8, x + 2y = 5$ I understand ...
0
votes
2answers
52 views

How to Show $M^2=7M-8I$ if $M$ is given in matrix form

$M$ is $2\times 2$ matrix, $m_{11}=3,\ m_{12}=-1,\ m_{21}=-4,\ m_{22}=4$ how to show $M^2=7M-8I$? we can only use substituting or trial and error method or got some more pro method..
0
votes
0answers
18 views

Reversing a rotation around an offset center of rotation

The best way to generally phrase my question is that I have a sphere offset from its center of rotation and a vector between the sphere and a target object at a known $(\theta,\phi)$ on the sphere. ...
0
votes
2answers
38 views

Eigenvectors and eigenvalues of matrices

Say that we have a square matrix $M$, and that a non-zero vector $v$ can be an eigenvector of $M$ if $Mv = kv$ for some real number $k$. This real number, $k$, can be called the eigenvalue of $v$ with ...
2
votes
1answer
36 views

What is the number of distinct elements in $S$?

Allow for these values: $$A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix} \text{ and } B = \begin{pmatrix} 1 ...