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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0
votes
0answers
11 views

Inverse kinematics - How do i compute the du?

I am at the moment trying to implement at jacobian based inverse kinematics solver, which is given a current homogeneous Transformation matrix r(q) and a desired homogenous tranformation matrix ...
2
votes
0answers
28 views

Matrix Calculus and Linear Transformations

I'm working on making the jump from differentiating real valued functions ($f: \mathbb{R}^n \rightarrow \mathbb{R}$) and vector valued functions ($g: \mathbb{R}^n \rightarrow \mathbb{R}^m$) to matrix ...
-3
votes
0answers
34 views

If I have matrix A, what is difference between $det(A), det(A_n), det(A_{n+1})$? [closed]

If I have matrix $A_n$, what is difference between $det(A_n), det(A_{n+1})$ and $det(A_{n+2})$? If somone wants to help and answer on question can that be with example?
0
votes
1answer
18 views

how do you solve the simultaneous equations 2x + y = 9 and x - 2y = -8 using the matrix method?

i first convert the bases into a 2x2 matrix and then i multiplied the inverse matrix by the 2x1 e/f matrix of 9 (on top) and -8 (on the bottom) which gave me x = 5.4 and y = 5 however the answer ...
1
vote
2answers
46 views

Nullspace and column space of invertible matrix

I want to show that the matrix $A$ $n\times n$ is invertible if and only if $N(A) = {0}$ and $C(A) = R^n$. So far, this is what I've got: Theorem: A is invertible $\implies N(A) = 0$ and $C(A) = 0$. ...
1
vote
2answers
30 views

Basis that contains a basis for a subspace

I have this exercise and I want to know if my answer is correct. The exercise is: Consider the linear space $\mathbb{R}^{2\times2}$ of $2\times2$ matrices with real entries. Consider $W$ contained ...
1
vote
2answers
72 views

How do I prove that $\det A_{n+2} = a \det A_{n+1} + b \det A_n$ for matrix $A$?

I have calculated: $\det A_1=2$, $\det A_2=3$, $\det A_3=4$, so I was putting some numbers in $\det A_{n+2} = a \det A_{n+1} + b\det A_n$ like $n=1$, $n=2$ ($\det$ $n\times n$ matrix) and get that ...
3
votes
3answers
72 views

$A^{2014}=0$ for a matrix A

Let A be a 3*3 matrix and $A^{2014}=0$. Must $A^3$ be the zero matrix? I can work out that I-A is invertible, but I don't know how to proceed further.
3
votes
1answer
50 views

Determinant of $P_n$

I am preparing for an exam on linear algebra within few days, so I am in desperate need for a solution for the following question: Question: Let $P_n$, $n\ge2$, be the $n\times n$ matrix whose ...
0
votes
0answers
17 views

Neural Net Matrix Multiplication

I'm trying to figure out the matrix multiplications for the implementation of a single hidden layer neural net for MNIST digit recognition. Like the following: ...
0
votes
0answers
18 views

making sense of this polar decomposition [closed]

I saw that the polar decomposition of $\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ is $\sqrt{\rho}\sqrt{\sigma}U$. $U$ is an unitary matrix. However, I cannot see how the $\rho^{1/2}$ can merge together. ...
0
votes
1answer
17 views

Can a Hermitian Matrix be Decomposed into a Sum of Unitary Matricies?

Given a Hermitian matrix $A$, when is it possible to write $A$ as a sum of unitary matricies as in the following form? $$ A = \sum_{i} a_i U$$ Where $U$ is unitary. Intuitively, because you have a ...
1
vote
1answer
19 views

Transition matrix of polynomial.

Good night, i need help with this. Find the transition matrix that goes from the basis W to the basis $\left\{ 2,1-2x,x^{2}-1,x^{3}-x^{2}+x\right\} $ I found a basis for W, $\left\{ ...
1
vote
1answer
55 views

How can we determine if the hyper-plane pass through the origin?

Let $A$ is an $n \times n$ matrix. Consider each row of $A$ as a point in $\mathbb{R}^n$; and assume these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point. The ...
-2
votes
1answer
32 views

Linear Algebra - Row echelon form [closed]

Find two different row echelon forms of: $$\left(\begin{matrix} 1&4\\ 3&11 \end{matrix}\right)$$ this exercise shows that a matrix can have multiple row echelon forms. I konw it is easy, ...
0
votes
0answers
17 views

Finite difference of radial Laplace operator doesn't give a symmetric (hermition in general) matrix

I'm using the central difference to convert the radial part of Laplace operator into a matrix. $\nabla^2 u = \frac{\partial^2 u}{\partial r^2}+$ $\frac{1}{r}$ $\frac{\partial u}{\partial r}$ which ...
0
votes
0answers
9 views

Determinants using Row Reduction replacement

I am aware replacement does not affect the value of determinant when doing a row reduction. However, I realised there isn't a good explanation on how to handle different forms of replacement when ...
1
vote
2answers
25 views

Expectation and variance of matrix valued random variable

Suppose I have a discrete matrix-valued random variable $X$, that is, I have defined a set of fixed matrices $\{Y_i\}_{i=1}^n$, and the random variable $X = Y_i$ with probability $\frac{1}{n}$. Is ...
0
votes
1answer
10 views

Derivative of a matrix times vector with respect to an entry of the matrix

I'm trying to find the derivative of $$\frac{\partial{A\vec{x}}}{\partial{A_{ij}}}$$ but I'm having some trouble figuring it out. $A\vec{x}$ will be a vector where the $i^{th}$ entry is $$A_{i1}x_1 + ...
1
vote
0answers
23 views

Eigenvalues of a tridiagonal block matrix

When a tridiagonal matrix is also Toeplitz, there is a simple closed-form solution for its eigenvalues, namely $ a + 2 \sqrt{bc} \, \cos(k \pi / {(n+1)})$ , for $ k=1,...,n$. Now my question is that ...
1
vote
2answers
18 views

matrix operations with transpose like properties

Has any one studied operations on matrices with transpose like properties - for example (A+B)' = A' + B' where ' stands for transpose. Also ' is its own inverse. Is there a common name for such ...
1
vote
0answers
35 views

Characteristic polynomial of matrix

If I wanted to find the eigenvalues of a matrix $\mathbf{A}$, then I could use these two options. $$\lambda\mathbf{I}-\mathbf{A}=\mathbf{0}$$ $$\mathbf{A}-\lambda\mathbf{I}=\mathbf{0}$$ However, ...
0
votes
0answers
12 views

the classes of similarity relation on matrices

we say two matrices $A,B∈M_n(F)$ similar if there exists a matrix $P$ that $$B=P^{-1}AP$$ Now suppose $F$ be an infinite field and assume the similarity relation and let $C$ be a finite class of ...
0
votes
1answer
31 views

eigenvalues and eigenvectors of 2x2 block matrix

My question is a really straightforward one: Is there an easier way to find the eigenvalues and/or eigenvectors of a 2x2 block diagonal matrix other than direct diagonalization of the whole matrix? $ ...
0
votes
3answers
34 views

Inverse matrix of sin and cos being multiplied.

find the inverse $$\begin{pmatrix} 5 e^{2 t} \sin(2 t) & 5 e^{3 t} \cos(2 t)\\ -6 e^{2 t} \cos(2 t)& 6 e^{3 t} \sin(2 t) \end{pmatrix}$$ I understand the inverse of $$\begin{pmatrix} ...
-1
votes
0answers
39 views

What do i need to construct a 3x3 rotation matrix? [closed]

What do i need to construct a 3x3 rotations matrix, that tells how a point should oriented to face object.
0
votes
1answer
30 views

Number of errors detected from a generator matrix

Consider the encoding function $\alpha : \mathbb{Z_2^2} \rightarrow \mathbb{Z_2^5} $ given by the Generator matrix $$ G = \begin{bmatrix}1&0 &1& 0& 0 \\0& 1 & 0 & 1 & ...
0
votes
1answer
25 views

Symmetric block matrix related

How to find eigenvalues of following symmetric matrix $\begin{bmatrix} kI-A & -A & -A & \cdots & -A\\ -A & kI-A & -A & \cdots & -A\\ -A & -A & kI-A & ...
0
votes
0answers
21 views

Find the value of $21D$ when elements of matrix are in H.P.

If $det$ represents and determinant and $$ \det\begin{bmatrix} a_1 & a_2 & a_3 \\ 5 & 4 & a_6 \\ a_7 & a_8 & a_9 \end{bmatrix}=D $$ and $a_1,a_2,a_3,5,4,a_6,a_7,a_8,a_9$ are ...
1
vote
0answers
17 views

Is it true that $\|(I+A)^{-1}\| \leq \|(I-B)^{-1}\|$ if $|A|<B$?

Is it true that $\|(I+A)^{-1}\|_\infty \leq \|(I-B)^{-1}\|_\infty$ if $|A|<B$? What if $A$ and $B$ are triangular matrices? Definition 1: $|A| = [|a_{ij}|]$, absolute value of every entry of $A$. ...
1
vote
1answer
28 views

Calculation of determinante via definition?

$\begin{pmatrix} -2 & 4 & 5 & -1 \\ 3 & 7 & 1 & -2\\ 0 & 8 & 2 & 3 \\ 1 & -1 & 3 & -1 \end{pmatrix} $ I have to calculate determinante via ...
1
vote
2answers
37 views

Prove the positive definiteness of Hilbert matrix

This is so called Hilbert matrix which is known as a poorly conditioned matrix. $$ A = \left(\begin{matrix} 1 & \frac{1}{2} & \frac{1}{3} & ... & \frac{1}{n} \\ ...
3
votes
3answers
170 views

Orthogonal matrix norm

If $H$ is an orthogonal matrix, then $||H||=1$ and $||HA||=||A||, \forall A$-matrix (such that we can writ $H \cdot A$). What norm is this about?
1
vote
1answer
66 views

Show that 1 + $\lambda$ is an eigenvalue of $I + A$

Show that if $\lambda$ is an eigenvalue of $A$, then 1+$\lambda$ is an eigenvalue of $I+A$. What is the corresponding eigenvector? What I have done so far (if it is correct at all...): ...
0
votes
2answers
30 views

How do I get the Rational Canonical Form from the minimal and characteristic polynomials?

Let's say I have the minimal polynomial and characteristic polynomial of a matrix and all its invariant factor compositions. How do I get a rational canonical form matrix from this?
1
vote
1answer
35 views

Solve $A^kx=b$ system using $LU$

I have the system $A^kx=b$ and the $LU$ factorization $A=LU$. How can I solve the system without actually calculating $A^k$?
0
votes
1answer
32 views

Different determinant for same matrix

I have the following matrix: $$ A=\begin{bmatrix} 2883,4675 & 44263,069125 & 724401,86824027 \\ 44263,069125 & 724401,86824027 & 12346864,4095603\\ 724401,86824027 & ...
1
vote
1answer
36 views

The rank of a positive square matrix $A$ for $A^k = A$

Determine the rank of $A$ if $A$ is a positive $n\times n$ matrix with $A^2 = A$. Give a geometric interpretation of $A$. What happens in the case when $A^k = A$ for some integer $k\geq 3$? ...
0
votes
0answers
26 views

Convert a 3d vector into a rotation matrix?

Is it possible to compute a Rotation matrix given a 3d vector given in the Euclidean space? and if not what would it need? An illustration of my situation. Illustration of my problem I have a ...
0
votes
0answers
32 views

Some sort of matrix.

How do you call this sort of algebraic entity as: \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} \end{bmatrix} I mean it's not a matrix as in a $m\times n$ array, ...
-2
votes
1answer
34 views

Eigenvalues of block matrix where blocks are related [closed]

How to find eigenvalues of following block matrix $A$ in terms of eigenvalues of matrix $B$? $A=\begin{bmatrix} 4I-B & -B \\ -B & 2I \\ \end{bmatrix}$ Where $B$ is square matrix of order ...
-1
votes
2answers
41 views

0 as approximate eigenvalue of a matrix [closed]

i got a problem that i cant solve. And i would be grateful for some help. Given the matrix $ X = \begin{pmatrix} 0 & 1 & & \\ -1 & 0 & \ddots & \\ & \ddots & \ddots ...
1
vote
1answer
38 views

Various matrix manipulations effect on determinant

Suppose that the matrix $A$ below has determinant $−3$. Find the determinants of $B$, $C$ and $D$. $$ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} ...
0
votes
0answers
25 views

How do i compute how much i can rotate my tool?

I am at moment trying to implement an Ball tracker for a robot arm with a stereo camera monted on it as its tool. Illustration: http://m.imgur.com/5oojXdh The camera provide me with an dx, dy, dz ...
2
votes
1answer
34 views

What Similarity of Matrices really mean?

If matrices are similar then what in layman language it would imply? Are there some properties of matrices which we would expect to be similar?
0
votes
0answers
20 views

How to find the characteristic polynomial for the following graph G

What is the closed form of characteristic polynomial (adjacency matrix) for the following graph $G$: With the help of eigenvectors, I found that $4$ eigenvalues of $G$ are that of $P_4$ and $6$ ...
0
votes
1answer
40 views

Matrix being equal

If $G = \left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} a,b\in \mathbb{R} : a \neq 0\right\}$ and assume $G$ is a group under matrix multiplication. Assume that $H = ...
3
votes
2answers
39 views

Solving generalized determinant related

How to solve following determinant by applying suitable elementary row/column transformations to obtain characteristic polynomial? \begin{align*} \left\vert \begin{matrix} -\lambda & 0 & 1 ...
2
votes
2answers
43 views

Finding an exponential matrix

How do I find the matrix exponential $e^{tA}$ with $$A = \left(\begin{matrix} 2 & 8 \\ 0 & 2\end{matrix}\right)$$ The eigenvalue is 2 with multiplicity 2, but it yields only 1 eigenvector ...
-1
votes
0answers
32 views

Find Singular value decomposition of Matrix [on hold]

Find the Singular Value Decomposition of the $3\times 2$ matrix: $$ \begin{pmatrix} 1 & 1 \\ -1 & 1 \\ 1 & 1 \end{pmatrix} $$ My book says the first step is to find the vector $v_1$ and ...