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
32 views

Equality in the Collatz-Wielandt-formula

Let A be a matrix with positive entries. The perron-frobenius-theorem states that A has a positive dominating simple eigenvalue, called the perron-frobenius-eigenvalue. I denote it with p(A). The ...
-2
votes
0answers
43 views

3D reconstruction from 2 images with baseline and single camera calibration

first posted: http://stackoverflow.com/q/24852151/3858076 i was forwarded to here cause this is more a mathematical problem so if anyone here could help me i would be very thankful. plz ignore the ...
0
votes
2answers
55 views

Suppose $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ lie on a plane through the origin.

What determinant is zero? What equation does this give for the plane? I need some help here, am pretty stuck
0
votes
1answer
355 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
1
vote
1answer
37 views
2
votes
1answer
44 views

Let $A$ be an $n \times n$ matrix with real entries. [closed]

Which of the following is correct? (a) if $A^2=0$, then $A$ is diagonalisable over complex numbers (b) if $A^2=I$, then $A$ is diagonalisable over real numbers (c) if $A^2=A$, then $A$ is ...
0
votes
2answers
46 views

Jordan chain when matrix has only one eigenvalue.

A $12\times 12$ matrix has sole eigenvalue $3$. It is given that the kernels of $A-3I$, $(A-3I)^{2}$, $(A-3I)^{3}$ and $(A-3I)^{4}$ have dimensions $4$, $7$, $9$ and $10$ respectively. What ...
0
votes
1answer
649 views

Diffrence between rotation and pure rotation

Hi i am trying to understand my teachers assignment. I have 2 write 2 Matlab functions ...
1
vote
1answer
28 views

Equality case in the Frobenius rank inequality

In many linear algebra books, the following rank inequalities are found: Frobenius inequality Let $A$, $B$ and $C$ be three matrices such that the product $ABC$ is defined. Then ...
1
vote
2answers
128 views

To Prove $x'Ax=\mathrm{tr}(xAx')=\mathrm{tr}(Axx')=\mathrm{tr}(xx'A)$

To prove, $x'Ax=\mathrm{tr}(xAx')=\mathrm{tr}(Axx')=\mathrm{tr}(xx'A)$ where  $A$ is a square matrix. $x'$ is the transpose of $x$. For each $x,x'$ are column vector, row vector.
3
votes
2answers
68 views

Matrix with all eigenvalues $0$ but not triangular?

Is the situation described in the title achievable? I am looking for a $3\times 3$ case specifically.
1
vote
1answer
568 views

Least square solution based on the pseudoinverse solved efficiently with singular value decomposition

Hi apologies it's hard to type out the problem, I have a lecture slide on neural networks. It says the fitting error gives the matrix: N by M matrix of thi's multiplied by Mx1 weights minus Nx1 ...
1
vote
1answer
27 views

Find the triangular matrix and determinant.

I have a 4x4 matrix and I want to find the triangular matrix (lower half entries are zero). $$A= \begin{bmatrix} 2 & -8 & 6 & 8\\ 3 & -9 & 5 & 10\\ -3 & 0 & 1 & ...
0
votes
2answers
20 views

what is the parametric function of the new Bezier curve?

The cubic Bezier curve can be given in matrix form as If a cubic Bezier curve is rotated by an angle 30 around x-axis what is the parametric function of the new Bezier curve?
1
vote
1answer
29 views

Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with. Can I claim ...
5
votes
1answer
62 views

Is this matrix positive-semidefinite in general?

for the matrix written below I was wondering if one can show that it is positive-semidefinite for $n>3$ and $0< \alpha<1$. (Or not. For $n=2, 3$ it works by showing that all principal minors ...
0
votes
0answers
22 views

Calculating Muliti-integral Time-Ordering operator

I am trying to solve the time-ordering operator Like this: $S(t)=\mathcal{T}e^{\int_0^tduA(u)\int_0^udvB(v)}S(0)$. How can I calculate S(t) with minimum error?
1
vote
2answers
519 views

Cayley Transform, Exponential Mapping and more…

Assume a self-adjoint operator, represented as hermitian matrix $H=H^\dagger$. To my knowledge there are at least 2 mappings of $H$ onto unitary matrices: Cayley's Transformation with ...
2
votes
1answer
66 views

asymptotic behaviour of coefficients in nonnegative matrix iteration

Let $A$ be a square matrix with nonnegative integer coefficients. Is there a simple way to prove that there is a "period" $d$ such that for all $0\leq r<d$, the coefficient $a_{i,j,n}$ at position ...
0
votes
0answers
21 views

Power iteration sequemce for a special nonnegative irreducible imprimitive matrix

Let $A \in \mathbb{R}^{n \times n}$ be nonnegative irreducible matrix with maximum positive eigenvalue equal to 1. Let's assume $A$ has $h$, $h > 1$ eigenvalues $\lambda_1, \dots, \lambda_h$ with ...
23
votes
1answer
709 views

Prove the determinant of this matrix

We have a square matrix, that all elements on main diagonal are zero, and other elements are following: $$a_{i,j}=\begin{cases} 1,&\text{if i+j belongs to Fibonacci numbers,}\\ 0,&\text{if ...
0
votes
2answers
61 views

Matrices: $AB=0 \implies A=0 \text{ or } \ B=0$

When A and B are square matrices of the same order, and O is the zero square matrix of the same order, prove or disprove:- $$AB=0 \implies A=0 \text{ or } \ B=0$$ I proved it as follows:- Assume $A ...
0
votes
1answer
28 views

How the Wronskian works

To prove linear independence of a set of functions, we say that given their Wronskian matrix W, Wx = 0 implies trivial solution (0,0,0,...) if the value (determinant) of the Wronskian is identically ...
1
vote
0answers
41 views

diagonal of pseudoinverse of laplacian matrix

I have to find the diagonal of the pseudoinverse of a laplacian matrix evaluated on a directed and weighted graph. My laplacian is defined as: L = D - A where: D is a diagonal matrix; Di,i the sum ...
0
votes
1answer
22 views

Linear Algebra - elimination and linear systems

By given this matrix: \begin{pmatrix}1&1&1&0\\2&3&k&1\\3&k&5&1\end{pmatrix} I need to find, what are the values of k the system has infinity/single/no solution. So ...
0
votes
1answer
22 views

What's the correct notation for a minimum of a row of a matrix?

Let I=1,...,m denote the indexes of the rows of a matrix A Let J=1,...,n denote the indexes of the columns of a matrix A Let xi,j denote the value of the element A[i,j] I need do use a notation to ...
1
vote
2answers
33 views

Matrix Algebra, Signs of solution

I have a system $AX = B$, where $A$, $B$ and $X$ are $N \times N$ matrices. I am interested in the properties of the solution $X$. $B$ has the following property: the diagonal terms are strictly ...
0
votes
0answers
25 views

Find the angle of rotation about a vector caused by application of a rotation matrix

I have a rotation matrix $R$ and a unit vector $\mathbf{v}$. How can I find the angle of rotation about $\mathbf{v}$ caused by the application of $R$?
0
votes
1answer
21 views

A question in matrix theory, SVD related.

For four $m\times n$ matrices A, B, A', B'. If $AA^\dagger=A'A'^\dagger, BB^\dagger=B'B'^\dagger$ and $AB^\dagger=A'B'^\dagger$, then if there always exists an unitary matrix V in U(n) such that ...
0
votes
0answers
54 views

QR decomposition algorithm

According to G. W. Stewart (Matrix Algorithms: Volume 1, Basic Decompositions) given an $n\times p$ matrix $A$, let $m=\min\{n,p\}$. The Stewart's Householder triangularization algorithm (Chapter 4, ...
0
votes
0answers
10 views

Matrix puzzle-Symmetric and skew symmetric matrices

Let P be an odd prime no. and T(P) be the following set of 2*2 matrices: T(p)={A=[a b] ; a,b,c belong to {0,1,2...,P-1} [c a] 1)Find the no. of A in T(P) such that A is either symmetric ...
2
votes
1answer
266 views

Column Space and SVD

I was reading Gilbert Strang's book and he says that if $A=USV'$ be the SVD of A ( assume square for the moment) then the nullspace of A is given by the last $n-r$ columns of V and the column space by ...
0
votes
0answers
20 views

Homography between known and unknown rectangle corners

I would like to know if there is a solution for the problem of homography estimation in the special case in which one of the views is unknown but has some constraints, particularly if we know the ...
8
votes
1answer
334 views

Is the zero matrix the only symmetric, nilpotent matrix with real values?

My intuition tells me that the zero matrix is the only matrix that is symmetric and nilpotent with real values, but I'm having trouble proving it (or finding a counterexample.) I have searched for ...
-1
votes
1answer
28 views

What's the solution of $F(R) = 0$? [closed]

$F(R) = -B + 2RYY^T + \mu R$ what is the solution to $F(R) = 0$? and why? Thanks
0
votes
1answer
55 views

Norm of a Vector

Suppose $A\inℝ^{n,n}$. We Define $$ \|A\| = \underset{\|x\| = 1}{\sup} \frac {\|Ax\|} {\|x\|}$$ Show it is a norm. Any thoughts?
3
votes
3answers
54 views

Geometric series of matrices

I am currently reading 'Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach' by J. Hubbard and B. Hubbard. In the first chapter, there is the proposition: Let A be a ...
0
votes
1answer
25 views

Find the matrix of T with respect to the standard basis in both the domain and the codomain

Let $T:\mathbb{R}^2→\mathbb{R}^2$ be given by $T((x,y)) = (x+2y , 3x-y)$ Question is what title states, Please help with detail.
3
votes
1answer
3k views

What is a “secular equation”?

Golub mentioned "secular equation" in his Matrix Computation and a slide. However I still don't get its definition. How is a secular equation defined? Thanks!
0
votes
1answer
296 views

Derivative (or differential) of symmetric square root of a matrix

Let A be a square, symmetric, positive-definite matrix. Let S be its symmetric square root found by a singular value decomposition. Let vech() be the half-vectorization operator. Is there a ...
2
votes
1answer
47 views

interger matrix whose square is identity

how can we find all the matrices with integer entries of size $n \times n$ such that $A^{2}=I$ and the matrix does not have fixed point in $\mathbb{Z}^n$ (except zero of course)? $-I$ is one example. ...
0
votes
2answers
34 views

Matrices Problem

I am doing the Cambridge O/L 2012 M/J P1 4024/12 Paper, Question number 12 (b). $$m = \begin{pmatrix} 3 \\ -2 \end{pmatrix}, \quad n = \begin{pmatrix} -1 \\ 4 \end{pmatrix} $$ Given that $$sm+3n = ...
-1
votes
2answers
47 views

Matrix with eigen values given find [closed]

Let$$ P=\begin{bmatrix} 0&-2&-3 \\ -1&1 &-1 \\ a&2 &b \end{bmatrix},$$ for some $a,b \in \mathbb{R}.$ Suppose that $1$ and $2$ are eigenvalues of $P$ and $$ ...
2
votes
1answer
30 views

Strange matrix multiplication behavior in Matlab

I noticed strange behavior of some calculations in Matlab. Matlab code listing: ...
0
votes
1answer
24 views

How do I calculate the Jacobian matrix of the transformation of a 1-m manifold to a chart (topology question)?

What I want to do is take a 1-m manifold (something like a circle), and transform a subset of that manifold into a chart. I want to represent that function from manifold to chart with a 1 x 1 matrix, ...
0
votes
1answer
30 views

what is the name of this matrix? does it have any special characteristics?

does anyone know the name of this matrix or if it has any special characteristics or how to calculate its inverse efficiently e.g. in a closed-form? [ \begin{array}{llllll} ...
1
vote
3answers
132 views

Show that these matrices are congruent.

Let $K$ be a field of characteristic$\ne 2$ and $u$ be an invertible element of $K$. Show that $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ and $\begin{pmatrix}u&0\\0&-u\end{pmatrix}$ are ...
0
votes
0answers
16 views

Matching two configurations by minimizing angles between pairs of points

I want to match two point configurations by rotation. The configurations are given by two $m$ by $n$ matrices $\boldsymbol A$ and $\boldsymbol B$ with each row representing a point in $\mathbb{R}^n$. ...
0
votes
1answer
1k views

Projection matrix onto null space

I have a matrix H and I want to find the projection matrix onto null space. How can I do this? Sorry if my question seems naive. Thank you, Tanja
4
votes
5answers
786 views

How to calculate gradient of $x^TAx$

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $ x^{T}Ax $ and he does that using the concept of exterior ...