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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1answer
16 views

From world space to object's space. Scaling.

I am developing a ray tracer and I need to compute intersections between many surfaces and rays. A classical method to make the computation time lower and the code simpler is to define some constants ...
1
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3answers
44 views

Problem with LU decomposition

I have this matrix: $$ A =\begin{bmatrix}1 & -2 & 3\\ 2 & -4 & 5 \\ 1 & 1 & 2\end{bmatrix} $$ After I decomposit it, I get: $$ L = \begin{bmatrix}1 & 0 & 0\\1 & 1 ...
0
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0answers
6 views

Inverse of symmetric block matrix with large amount of blocks

I need to invert a large block matrix, where each individual block is small and of fixed size. e.g., A 1000*1000 matrix where each block is of size 10*10, etc. I could only find formulas for 2*2 block ...
0
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0answers
20 views

Fixed points and permutations.

Let $\psi ,\varphi \in {S_n}$ two permutations. Let $M$ a matrix such that $a_{i,j}=1$ iff $i=\sigma(j)$ where $\sigma \in S_n$ ($0$, otherwise) I already showed that $tr(M) = \left| {\left\{ {k \in ...
0
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1answer
43 views

How do I multiple these matrices together?

As a personal brain exercise, I've recently been trying to work out the math involved with rotating vertices around an arbitrary axis in 3D space. To do so, I've been relying very heavily on the ...
7
votes
1answer
63 views

What property of a matrix causes $\|e^{tA}\|_2$ to oscillate as $t\rightarrow\infty$?

What property of a matrix causes $\|e^{tA}\|_2$ to oscillate as $t\rightarrow\infty$? The best I can come up with is that $A=bi\cdot M$ for $b$ a non-zero real number and $M$ a non-zero idempotent ...
1
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0answers
24 views

Matrix exponentiation by Lagrange interpolation formula

Okay so I'm trying to get my head around calculating powers and functions of matrices. So given a matrix $A=PJP^{-1}$ suppose we wish to calculate $A^n$ using the Lagrange interpolation formula as ...
1
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1answer
33 views

Zeroing out one element of a correlation matrix

Let $R$ be a correlation matrix. Is there some $T$ such that $$ TRT' = Z^{(ij)}, $$ for $i \neq j$, where the $(m,n)$th element of $Z^{(ij)}$ is given by $$ Z_{mn}^{(ij)} = \begin{cases} ...
0
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0answers
75 views

Matrix graph and irreducibility

How do I prove that if $A\in\mathbb C^{n\times n}$ is a matrix then it is irreducible if and only if its associated graph (defined as at Graph of a matrix) is strongly connected?
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0answers
17 views

What are incoherent matrices

What does incoherence means in terms of matrices? I am brushing on some compressive sampling theory and I did not find any easy to understand or straight forward answer about what does the word ...
0
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0answers
11 views

Examples/online resources for Givens rotation

I want to write a program to implement the Givens rotation. All the online resources I found explain the concept with just 2x2 matrix. I wanted to know the mathematical formula for 3x3 or nxn matrix ...
3
votes
3answers
82 views

Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant,…but they are not similar.

Question: Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant, trace and characteristic polynomial, but they are not similar to each other. My thought: I come up with two ...
4
votes
1answer
112 views

Why do we define addition of matrices only when they have the same size

What happens if we define $$ \begin{pmatrix} 1 & 2 \\ 1 & 2 \\ 1 & 2 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 ...
1
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2answers
31 views

reduction of a skew-symmetric matrix

Birkoff and MacLane state that any real symmetric matrix $A$ has the form $ A = P^{-1}BP $ where $ B^2 $ is diagonal and they ask for a proof as an exercise. It seems to me that if $A$ is ...
0
votes
1answer
33 views

Householder matrix Uw acts as the identity on the subspace w

How can i show that a Householder matrix $U_w$ acts as the identity on the subspace $w$? and that it acts as a reflection on the one-dimensional subspace spanned by w; i.e., $U_w(x) = x$ if $x$ is ...
1
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0answers
47 views

Can I calculate this sum using matrix multiplication?

I would like the following sum for the matrices $A,B$, both $n \times n$: \begin{align} \sum(A_{ij} \cdot B_{jk} \cdot A_{kz} \cdot B_{zf}) & \text{ for all } j,k,z,f \text{ such that } ...
0
votes
1answer
72 views

Peculiar Matrix

I came up with this idea recently and I want to go deeper in this, but it has been difficult to me. Hope someone can help me on this. Suppose I have a matrix of order $(n^2-1)\times (n^2-1)$ with ...
0
votes
2answers
20 views

Linear maps, matrices, nullity and rank.

I am currently trying to solve this question in my first year linear algebra course: I understand that the assoc. matrix is the coefficients, eg for (a) [[1, -1];[5, 0]], but I'm not sure how to ...
0
votes
0answers
26 views

Find basis when there is only one parameter?

Find a basis of {(1,-1,1), (2,0,1), (1,1,0)}. I set up the matrix and RREF'd it, and I got that: x1 = t x2 = -t x3 = t So I set t = 1 and find one vector (1,-1,1). This is half of the right ...
1
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1answer
39 views

Linear Algebra True/Flase

Are these two statements true or false, if brief justification/counterexample could be given it would be appreciated. $(1)$ $I_{V}$ is an identity operator on vector space $V$, $\dim V=n$ and $A$ is a ...
1
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1answer
20 views

Rank of matrix a submatrix $B$ from $A$

Question: A submatrix $B$ consisting of "s" rows of $A$ is selected from an n-square matrix $A$ of rank $r_{A}$. prove that the rank of $B$ is equal to or greater than $r_{A}+s-n$. My thoughts: I ...
1
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1answer
28 views

$\frac{d(X'X)}{dX}=?$

Thanks a lot for reading my thread. I am wondering what is the derivative of $X'X$ with respect to $X$? Here $X$ is a vector/matrix, and $X'$ is the Hermitian matrix of $X$; It would be great if ...
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votes
0answers
53 views

matrix…urgent solving required [closed]

Consider the linear equations where, \begin{cases} 2x-y+z=a\\ x+y+2z=b\\ 3y+3z=c \end{cases} Find the condition on $a$, $b$ and $c$ such that the system of linear equations has (i) no solution (ii) ...
0
votes
3answers
38 views

How to find rank of a matrices?

Here is the question given in my text book IF the rank of the matrix $\begin{bmatrix}-1 & 2 & 5\\2 & -4&a-4\\1&-2&a+1\end{bmatrix}$ is 1, then the value of a is: a) ...
1
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1answer
26 views

Is there a pseudo inverse $X$ such that $ABX=A$?

Question The title pretty much sums it up. I need to find a matrix $X$ such that: $A B X = A$, with $A\in R^{n\times n}$, $\text{rank}(A)=n$, $B\in \mathbb{R}^{n\times m}$ given. The matrix $X$ ...
1
vote
1answer
28 views

pseudo-inverse by SVD decomposition has not accurate results?

The goal is finding $\frac{{\partial f}}{{\partial {\bf{A}}}} = 0$ where $ f\left( {{\bf{A}},{\boldsymbol{\alpha }}} \right) = {\left( {{{\bf{p}}^{\bf{T}}}{{\bf{A}}^{\bf{T}}}{\boldsymbol{\alpha }} ...
1
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0answers
31 views

Is there would be any matrices $A$ & $B$ where the relation $AB$-$BA$=$I$ holds? [duplicate]

Here , the actual question is to find any matrices $A$ and $B$ such that $AB-BA=I$ relation holds. but actually, I dont think that we could not find any such matrices. As, the diagonal elements of ...
0
votes
1answer
13 views

Obtaining consistent triangle surface normals.

I am given 3 points in a random order like so... calculateSurfaceNormal(point1, point2, point3); I have implemented the method by simply saying... ...
-2
votes
0answers
31 views

Matrix of Expectation of Random variables Update [closed]

I am not a math guy, but here I have encounter a problem about finding an inverse matrix, which the original matrix are elements of expectation of random variables. I think it is an optimization ...
1
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1answer
18 views

How to prove that the rank and the nullity of similarity invariants are the same?

If given matrix A and P-1AP How do prove that the rank(A) and rank(P-1AP) are the same? Also how do you prove that the nullity(A) and nullity(P-1AP) are the same?
0
votes
3answers
39 views

Calculate Matrix A from eigenvalues, but no given eigenvectors

Here is my question: Write down a nontriangular 3 by 3 matrix whose eigenvalues are 6, 9, 2. I understand that you can calulate Matrix A using the formula A=V$\Lambda$$V^-1$, but is there a way to ...
1
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1answer
27 views

Matrix of a given operator $A \otimes A$

Let $V$ be a 3-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $e_{1}$, $e_{2}$, $e_{3}$ ...
1
vote
1answer
119 views

Graph of a matrix

How to define the graph of a square matrix $\mathbf{G}$ with real entries? I know that given a graph $\Gamma(V, E)$, one can define its adjacency matrix $\mathbf{A}$. But given a matrix $\mathbf{G}$ ...
3
votes
1answer
40 views

Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as with SVD?

Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as using SVD?
0
votes
2answers
32 views

Similar matrix and diagonal matrix

What is the difference between a similar matrix and a diagonal matrix? According to my textbook, the definition for both is basically: B=P$^{-1}$AP. Say if there are three matrices: A, B and C. If A ...
-1
votes
1answer
60 views

How to find linearly independent columns in a matrix

For a general square matrix $A$, how do I find which columns are linearly dependent? When I say linear independent I mean not linearly dependent with any other column or any combination of other ...
1
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0answers
26 views

How's the damping factor in Google PageRank algorithm calculated

I'm doing some researches about Google's PageRank algorithm for my thesis, I've found that the damping factor x (for example), where x is in : P` = x.P + (1-x)Q where P is the original ...
1
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2answers
121 views

Show that the order of the matrices must be even.

Let $A,B$, two matrices with the order of $n\times n$. Given that $AB + BA = 0$ and $A,B$ are invertible (meaning, there are $A^{-1}, B^{-1}$). Prove that $n$ must be even number. $$\eqalign{ ...
5
votes
4answers
301 views

Mysterious Proof about Induced Norms (was: Uniqueness of SVD)

In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen), argues as follows: Let $\sigma$ be the first singular value of A, and $v_{1}$ the ...
0
votes
0answers
17 views

Gaussian matrix khatri rao product property

if $A=[a_1,a_2,...,a_n]$ is a random matrix of $m \times n$ and $m < n < m^2$, its elements are independent Gaussian random variable,let $B=[a_1⊗a_1,a_2⊗a_2...,a_n⊗a_n]$. $\otimes$ means ...
0
votes
0answers
56 views

Transform one curve into another

I have been working on something for a while now, and I can't really get my head around it. I consider two curves with data points and want to determine the most optimal transform from one to another. ...
0
votes
0answers
10 views

Derivation of a fixed effects estimator

I've come across parts of a derivation of an estimator in a paper i don't understand. The log likelihood function is where $Y_i=(Y_{i1},...,Y_{iT})'$, and $X_i$ is a $T\times k$ matrix ...
0
votes
1answer
21 views

How to find the rotation matrix that will align an arbitrary vector to an axis

If I have a vector that starts at the origin, how can I find the transformation matrix that will align it with the positive y-axis. So it basically turns into a positive-y axis? EDIT: I also forgot ...
1
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1answer
33 views

To prove that these matrices are invertible

Let $A$ and $B$ be $n \times n$ matrices such that $||I - AB|| < 1$. Prove that $A$ and $B$ are invertible, and $$A^{-1} = B \sum\limits_{k=0}^{\infty} (I - AB)^k \text{ and } B^{-1} = ...
2
votes
1answer
40 views

How do you handle this kind of probability?

What is the probability of selecting a singular matrix from $\Bbb{R}^{3\times 3}$? I calculated it to be zero based on their being approximately $9$ degrees of freedom to choose entries of $A$ such ...
1
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0answers
33 views

Matrices of the form $A^p=(a_{ij}^p)$

I am wondering if there is a name for these kind of matrices and if they are interesting or not? Do they even exist? Let $A$ be a $n\times n$ matrix with elements $a_{ij}$. $A= (a_{ij})_{i,j\in\{1, ...
1
vote
2answers
42 views

How many $3 \times 3$ matrices are singluar?

How many $3 \times 3$ matrices are singluar? Describe the methodology used to achieve the result.
0
votes
3answers
37 views

Find the conditions required for the values of a, b, and c that make the following matrix symmetric.

Set up the system: $$A = \begin{bmatrix} 5& a+b+c& a-b \\ 3& -7& 2\\ 1& a+c & 6 \end{bmatrix}$$ I did it like this: ...
0
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1answer
25 views

Methods for solving linear systems

This is such a basic topic but there are so many different methods proposed for solving a linear system of equations. I recently found a very good source but couldn't really make sense of all the ...
1
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0answers
27 views

What does it mean to compute a normal to a triangle in a “clockwise direction”

I am trying to understand how this works. I am given 3 points, each representing a vertex of a triangle. I must then "organise" the points and calculate the normal of the resulting triangle in a ...