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
1answer
67 views

A matrix as a point in $\mathbb{R}^{nm}$

I just had a really quick question to ask. I was reading a book on linear algebra and have just been trying to wrap my head around what exactly a matrix represents. At one point, the book said "In a ...
3
votes
3answers
47 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 ...
2
votes
3answers
72 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 ...
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
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 ...
-6
votes
0answers
46 views

Show that any linear transformation maps the origin to the origin. [closed]

Please solve this question. Show an example to prove it.
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 ...
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
0answers
32 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 ...
-1
votes
1answer
46 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 ...
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 ...
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
35 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 ...
2
votes
2answers
17 views

Generate an integer matrix such that all submatrices are non-singular

I need to generate an $\infty \times N$ integer matrix with a few properties. The top $N$ rows (and $N$ columns) should be the identity matrix. Any square submatrix (meaning the result after ...
0
votes
0answers
31 views

Square matrix whose sum of squared elements equals 1.

I'm doing some applied work where I've come across examples that involve real valued square matrices that hold the following property, which expressed using tensor notation is $$A_{ij}A_{ij} = 1$$ ...
0
votes
0answers
14 views

Time derivative of rotation matrix R is the product of a skew matrix and R

Can someone please give a proof that if $R(t)$ is a rotation matrix function of time, then its derivative at time t is equal is equal to a skew matrix times R(t). Thanks
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. ...
4
votes
1answer
129 views

If $AA^*=AA$, how to prove $A$ is an Hermitian? [duplicate]

If $A$ is an $n \times n$ matrix and $AA^*=AA,$ how to prove $A$ is Hermitian?
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
22 views

Finding a jordan basis for a jordan form

Let $$A = \left(\begin{array}{cccc} 1&0&0&0\\3&-2&0&0\\14&0&-2&0\\8&-1&1&-2 \end{array}\right)$$ Easy to verify that $f_(x) = (x-1)(x+2)^3$. So the ...
0
votes
0answers
32 views

A question in matrix polynomial

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
-1
votes
0answers
18 views

How do I parametrize the solution space of this system of equations? [closed]

\begin{pmatrix} 1& 2& 4& 0& -3& |&-1\\ 0 &0& 0& 1 &-4 &|& 1\\ 0 &0 &0 &0 & 0 &|& 0\\ 0 &0& 0& 0 & 0 &|& ...
0
votes
3answers
64 views

Given $A^2-4A+I=0$, show that $ A^3=15A-4I$

If have a question like this , can we using equation method or deduction method to answer the question?? Or we need to answer the question by substituting the matrix??
0
votes
0answers
58 views

Needs an explanation on why I obtain this covariance matrix

Let's say $n$ is an even integer. I'm playing with a column vector $\mathbf{v}$ which must satisfy the following three requirements: It's a length-$n$ vector of +1s and -1s. It has the same number ...
0
votes
0answers
25 views

quaternions are less versatile than matrix?

I am doing some research looking should I implement quaternions or matrices. What I've seem to come across is that while quaternions can be better for doing smooth rotations and dual quaternions can ...
0
votes
1answer
24 views

Matrix,Linear algebra,polynomial,finite field,notation

In the book by Arora and Barak,Computational Complexity,on page 168,1st paragraph, there is a notation which I do not understand. They write For every $n \times n$ matrix $A$,and $i\in [n]$,we define ...
1
vote
1answer
41 views

Finding the Jordan Form and basis

$$A= \begin{pmatrix} 2&1&2\\ -1&0&2 \\ 0&0&1 \end{pmatrix}$$ I found that $$f_A(x)=m_A(x) = (x-1)^3.$$ So the Jordan form must be: $$J= \begin{pmatrix} 1&0&0\\ ...
0
votes
3answers
83 views

Prove or disprove: $A=A^\top \land B = B^\top \Rightarrow AB = (AB)^\top$

where $A,B\in\mathbb{R}^{n\times n}$. My current solution is that this will only work iff $A$ and $B$ commute. Since: $(AB)^\top = B^\top A^\top = B A$ $\ $ ($=AB$. iff $A$ and $B$ commute.) I ...
0
votes
1answer
35 views

Are there necessary and sufficient conditions on $A$ and $B$ such that each row of $AB$ has a nonzero entry?

Let $A$ be an $n_A \times n$ matrix and $B$ an $n \times n_B$ matrix. What are necessary and sufficient conditions on $A$ and $B$ such that each row of $AB$ has a nonzero entry?
0
votes
0answers
28 views

Article writing: How to represent a matrix by its elements?

Intuitively I guess that parantheses with subscript and superscripts is a way of representing a matrix or an array by its elements, e.g., $$ A = (a_{i,j})_{i,j=1}^n $$ (This is taken from here) ...
2
votes
2answers
45 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 ...
0
votes
0answers
25 views

Increasing Matrix

Consider real matrix-vector multiplication. I am just curious whether there exists a $p\times p$ matrix $A$ such that if $x$ is a $p\times 1$ real vector whose entry is in ascending order, i.e., $x_1 ...
1
vote
4answers
59 views

Find all $2 \times 2$ matrices $A$ and $B$ such that $AB = BA$

Find all possible $2 \times 2$ matrices A that for any $2 \times 2$ matrix B, AB = BA. Hint: AB = BA must hold for all B. Try matrices B that have lots of zero entries. I'm clueless as to how to ...
1
vote
3answers
56 views

If $\mathbf{A}$ is a $2\times 2$ matrix that satisfies $\mathbf{A}^2 - 4\mathbf{A} - 7\mathbf{I} = \mathbf{0}$, then $\mathbf{A}$ is invertible

$\mathbf{A}$ is a $2\times 2$ matrix which satisfies $\mathbf{A}^2 - 4\mathbf{A} - 7\mathbf{I} = \mathbf{0}$, where $\mathbf{I}$ is the $2\times 2$ identity matrix. Prove that $\mathbf{A}$ is ...
0
votes
2answers
19 views

Determine if a basis consists of eigenvectors

So, this might be a silly question, but here it is. I am doing a couple of problems computing $[T]_\beta$, and determining whether $\beta$ is a basis consisting of eigenvectors of $T$. My problem is ...
-4
votes
1answer
38 views

prove Determinants are equal [closed]

given matrix $A$, matrix $B$ is the matrix we get after adding row $i$ plus a linear combination of other rows to the row $i$, in matrix $A$. Prove that : $detA=detB$
1
vote
0answers
18 views

Finding points inside of a box

I have a set of points in 3D that define a large, complex object. These points are rendered in OpenGL for an Android app that I am programming. In this app, the user translates the center of the box ...
0
votes
0answers
17 views

When to use which condition number? (which norm)?

The condition number is used to determine how sensitive b is to changes in A in the equation ...
-1
votes
1answer
52 views

Calculating determinant of matrix $n\times n$ [closed]

matrix M $n\times n$ given, which its left-right diagonal contains the numbers from $1$ to $n$, and all the other numbers equal $n$. calculate the determinant of the matrix M.
6
votes
0answers
62 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
1answer
35 views

Linear Recurrence Using matrix exponentiation

Matrix Exponentiation can be used to solve Linear Recurrence . I know how to solve linear recurrences like : $f(n) = f(n-k_1) + f(n-k_2) + ... +\ constant$ But i couldn't find any information on how ...
1
vote
3answers
102 views

Eigenvectors of real symmetric matrices are orthogonal (more discussion)

This is an old question, and the proof is here The proof assumed different eigenvalues with different eigenvectors. My question is how about the repeated root? How to guarantee there will not have ...
3
votes
1answer
18 views

Get amount of submatrixes from $a \times b $matrix

I was trying to do the following exercise Given a grid of size $a \times b$, write a formula able t calculate the total number of rectangles contained in this rectangle. All integer sizes and ...
0
votes
0answers
32 views

What is the minimizer of the matrix norm and it's significance?

For $M_{n\times n}$ a p.s.d real matrix, if we minimize $||M^{\frac{1}{2}}x||_2$ over $x$ under a linear constraint on $x$ as in $Ax=b$, where $b$ is non-zero. what is the significance of this $x$? ...
0
votes
1answer
61 views

eigenvalues and $A^k$

Consider $$A=\begin{pmatrix} s+1 & 1-t \\ -1-t & s-1 \end{pmatrix}$$ where $s$ and $t$ are real numbers. (a) For which values of $s,t$, $A$ is not diagonalizable. (b) For which choice of ...
0
votes
0answers
22 views

Matrix Multiplication By Rows And Columns

A given matrices $A$ and $B$ produces $A\cdot B=C$. We can say that: $Col(C)=$ linear combinations of the $Col(A)$ $Rows(C)=$ linear combinations of the $Rows(B)$ And $A\cdot[B^1 ...
2
votes
1answer
29 views

Convergence of a sequence of $2\times 2$ real matrices

My Try: So $a_n$ can be written as a series very similar to the taylor series of sin: $\displaystyle a_n=\sum_{k=0}^n \frac{(-1)^k b_k}{(2k+1)!}$ for some $b_k$ to be determined. But it is very ...
0
votes
1answer
13 views

Proof of some properties about orthogonal matrices

$Q$ is an orthogonal matrix, how to prove. $$\langle Qu,Qv\times Qw\rangle=\langle u,v\times w\rangle$$ for any $u,v,w$ which belong to $\mathbb R^3$ Much obliged if you can help me!
3
votes
3answers
41 views

Proving whether linearly independent [closed]

I've been working on this for almost half hour, can someone answer this question perhaps? Thanks. Let {$\vec u_1, \vec u_2, \ldots , \vec u_k$} be a linearly independent set of vectors in ...
0
votes
1answer
24 views

Common solutions to quadratic equations associated to self-adjoint matrices

Let $\mathcal{H}$ be a complex Hilbert space of dimension $d<+\infty$, and let $\{|n\rangle\}$ with $n=0,\cdots,d$ be an orthonormal basis in $\mathcal{H}$. Let $\mathbf{A}$ be a self-adjoint ...