1
vote
0answers
11 views

Multidimensional fitting of two data sets

My problem is the following: A laser gives out a bunch of data points which are reflected off a metal surface and recorded by a camera attached to the side of the laser. The image the camera receives ...
1
vote
0answers
36 views

Eigenvalue formula for 4x4 symmetric matrix

Is there a formula/algorithm that is accurate to used in finite precision arithmetic (aka numerical stable ) for small symmetric matrix of size 4x4. Additionally I'm looking if it require similar ...
2
votes
2answers
33 views

If the 2-norm of a matrix is small, the trace of the matrix is also small

Is it true that If the 2-norm of a symmetric real matrix is small, then the trace of the matrix is also small? I played around with some matrices in MATLAB and discovered this phenomenon. Does there ...
1
vote
1answer
46 views

How to solve the equation $AX=B$ in Matlab?

I am trying to solve an equation of the form AX=B where A, X and B are following matrices. I have the A and B matrices and I have to find the value of matrix X. How can I find the value of matrix X. I ...
0
votes
1answer
14 views

Prove that the direct sum of a symmetric and skew symmetric matrix belongs to $M_n(K)$ using $A_{ij}$ and $A_{ji}$ notation.

Basically Let $M_n(K)$ be an $n\times n$ matrix of a $K$ vector space. $U =\{A\in M_n(K)\;|\;A_{ij}=A_{ji}\}$ $W =\{A\in M_n(K)\;|\;A_{ij}=−A_{ji}\}$ So I don't understand my mark scheme. It says ...
0
votes
1answer
36 views

First-order linear differential equation for matrix valued functions of size $3\times 3$

I have two matries given by (M' means derivative w.r.t x) $ M=\left( \begin{array}{ccc} f_1(x) & f_2(x) & f_3(x) \\ f_4(x) & f_5(x)& f_6(x) \\ f_7(x) & f_8(x) & ...
3
votes
1answer
42 views

Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
1
vote
2answers
39 views

Show that 2 matrices belong to a square matrix by taking the transpose. Vector spaces

Let $M_n(K)$ be an $n\times n$ matrix of a K vector space. \begin{align} U &= \{A ∈ M_n(K) | A_{ij} = A_{ji} \} \\ W &= \{A ∈ M_n(K) | A_{ij} = -A_{ji} \} \end{align} Prove that $U$ and $W$ ...
0
votes
0answers
17 views

Recursion on Matrix

We have a given matrix recurrence given, $ (\curlyvee_i,\curlyvee_{i-1})_{1\times3}= (\curlyvee_{i-2},\curlyvee_{i-3})_{1\times3}{\begin{bmatrix}A_{i-1}A_i+B_i & A_{i-1} \\B_{i-1}A_i & ...
2
votes
2answers
34 views

$m \times n$ matrix where $m < n$

So I'm a long distance student and I need some help to bounce ideas off of other people who understand the work. Fellow students are few and far between. So while this is an assignment question, I ...
1
vote
6answers
114 views

How to raise a matrix to the power of $13$ without boring, repetitive multiplication?

how can i show $\begin{pmatrix}1 & 1 & 1 \\0 & 1 & 1 \\0 & 0 & 1\end{pmatrix}^{13}=\begin{pmatrix}1 & 13 & 91 \\0 & 1 & 13 \\0 & 0 & ...
0
votes
1answer
22 views

Pseudoinverse and orthogonal projection

Given the matrix $A= \begin {pmatrix} 1 & 1 &1 \\ -1 & 1 & 0 \\ 0 & 2 &1 \end{pmatrix}$. (i) Determine the orthogonal projection $p:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ on ...
2
votes
2answers
59 views

a question about rank of a matrix

Suppose $A$ is a $m\times n$ matrix. Show that $\mbox{rank}\,A=m$ if and only if there exists a $n\times m$ matrix $B$ such that $AB=I_m$. I have proved the case $AB=I_m$ eventuates ...
0
votes
0answers
14 views

What happens when scaling a rectangle using a pivot point?

With multitouch screens, you can pinch to zoom. When such a gesture is triggered you are supplied with: An x scale factor; A y scale factor; A x pivot point; A y pivot point. When I have a ...
1
vote
0answers
37 views

$tr(A)=0$ then exsists $P,Q$ such that $A=PQ-QP$ .

Let $\mathbb{F}$ be an arbitrary field and $A\in M_{n\times n}(\mathbb{F})$ such that $$tr(A)=0$$ Now show that there exists $P$,$Q$ $\in M_{n\times n}(\mathbb{F})$ such that $$A=PQ-QP$$ It is ...
2
votes
1answer
13 views

Matrix with respect to basis.

Define D:$\wp_{2}$($\mathbb{R}$) $\mapsto$$\wp_{2}$($\mathbb{R}$) by $D(p)(x) = p'(x)$ , Find the matrix of $D$ with respect to the basis $\{1, 1+x, 1+x+x^2 \}$ I was thinking this would be ...
2
votes
1answer
27 views

Elementary Matrix and row of zeros

If you have the following matrix can $k$ be any number? \begin{pmatrix} 1 & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & 1 \end{pmatrix} So this is obviously an assignment question, ...
1
vote
0answers
21 views

How to find the rank of a toeplitz matrix?

Is there any trick to compute or estimate the rank of a toeplitz matrix ? Or is this still unknown for a general toeplitz matrix ?
3
votes
1answer
67 views

Is it true that $u + v$ is an eigenvector corresponding to the eigenvalue $\lambda$?

Let $A$ be an $n \times n$ matrix, and $u, v$ be eigenvectors corresponding to an eigenvalue $\lambda$ of $ A$ (that is, $Au = \lambda u$ and $Av = \lambda v$). Is it true that $u + v$ is an ...
0
votes
0answers
34 views

Tree Traversal - Simple Puzzle type Issue.

This is a puzzle like question,based on Fibonacci like structure of the tree. Actually it is a short question with out any complex concepts. It appears bit big,since I have added explanations with ...
1
vote
1answer
67 views

Maximum determinant of a $m\times m$ - matrix with entries $1..n$

I want to find the maximal possible determinant of a $ m\times m$ - matrix A with entries $1..n$. Conjecture 1 : The maximum possible determinant can be achieved by a matrix only ...
0
votes
2answers
49 views

Characteristic polynomial and eigenvalues of a $3 \times3$ matrix.

Hi so I have to find the characteristic polynomials and the eigenvalues of the matrix: $$A = \begin{bmatrix}1 & 0 & 3\\2 & -2 & 2\\3 & 0 & 1\end{bmatrix}$$ So I know you use ...
0
votes
1answer
75 views

Characterize matrices A such that trace(AC)=0 for every matrix C with trace(C)=0

$A$ is an $n\times n$ matrix on the field $F$ such that for every $n\times n$ matrix $C$ with $\operatorname{trace}(C)=0$ we have $\operatorname{trace}(AC)=0$. Can we characterize such matrices $A $? ...
0
votes
0answers
22 views

Which casses of matrices contain A and which contain B? Linear Algebra

Am pretty confused about classes, I don't know what it means, so so I can't really do part_A and I need your help with it? For part B, I got all eigen = 1 for matrix A, and 0 for matrix B, Is this ...
0
votes
1answer
19 views

How do you get nullspace N(A) to be orthogonal to C(A^H)

In the picture below, C(A) is given in number7, but I am doing number_8. Ii did a gauss jordan where by i subtracted R2-iR1 to get 0 belo 1st pivot and 1 as the second pivot in column2, row2. Then I ...
0
votes
3answers
78 views

Orthogonal diagonalization of Symmetic Matrices

Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial \Delta (t). Step 2: find the eigenvalues of A which are the roots of \Delta (t). Step 3: for each ...
3
votes
5answers
579 views

Are inverse matrices unique?

Does a matrix have only one inverse matrix (like the inverse of an element in a field)? If so, does this mean that $A,B \text{ have the same inverse matrix} \iff A=B$?
3
votes
1answer
64 views

Surprising necessary condition for a “shift-invariant” determinant

Let $A$ be a $4\ x\ 4$ binary matrix and $Z=\pmatrix {s&s&s&s \\ s&s&s&s \\s&s&s&s \\s&s&s&s}$ Then $\det(A+Z)=\det(A)=1\ $ (independent of s, so ...
-3
votes
0answers
66 views

Series expansion - Involving matrix exponent [closed]

I have a skew symmetric matrix $$C=\left( \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \\ \end{array} \right).$$ Given Conditions $x = ...
0
votes
1answer
28 views

Diagonalization of Skew symmetric matrix

I have a skew symmetric matrix $$C=\left( \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \\ \end{array} \right).$$. and we have the relation $C=UDU^{-1} ...
1
vote
1answer
92 views

Simple proof that a $3\times 3$-matrix with entries $s$ or $s+1$ cannot have determinant $\pm 1$, if $s>1$.

Let $s>1$ and $A$ be a $3\times 3$ matrix with entries $s$ or $s+1$. Then $\det(A)\ne \pm 1$. The determinant has the form $as+b$ with integers $a$,$b$ and it has to be proven that $a>0$ if ...
2
votes
1answer
32 views

Determinant of a matrix shifted by m

Let $A$ be an $n\times n$ matrix and $Z$ be the $n\times n$ matrix, whose entries are all $m$. Let $S$ be the sum of all the adjoints of $A$. Then my conjecture is $\det(A+Z)=\det(A)+Sm$ , in ...
0
votes
2answers
48 views

How are signs on eigen vectors chosen, am confused? Linear Algebra

I have found the eigen vaues, I also know that you can find the eigenvectors through a Gausian Jordan. -- x1, gauss jordan gives me rows(1 -1/3 ,, 0 0 ), so [a, b] = [1,3] For vector x2, GJ gives ...
0
votes
2answers
75 views

Square root of symmetric matrix

I have a symmetric matrix $A$. How do I compute a matrix $B$ such that $B^tB=A$ where $B^t$ is the transpose of $B$. I cannot figure out if this is at all related to the square root of $A$. How to ...
0
votes
0answers
25 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
0
votes
1answer
30 views

Linear Transforms & Matrices

$T:R^4 -> R^3$ Linear Transform This matrix is $[T]_{B2}^{B1}$ = A =\begin{pmatrix}1&2&3&4\\1&4&0&2\\2&2&9&10\end{pmatrix} After elimination we get: ...
0
votes
1answer
30 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
0
votes
0answers
37 views

proof that T^k is a positive operator

so the book (Axler linear algebra done right) asks me to prove that if $T$ is a positive operator then $T^k$ is also positive , now the book defines a positive operator as an operator which is self ...
0
votes
1answer
43 views

The geometric multiplicity

By given this matrix: \begin{pmatrix}0&a&0\\0&0&1\\0&0&0\end{pmatrix} Why for any a which is not 0 the geometric multiplicity = 1? and why for a = 0 the g.m. = 2? I don't ...
4
votes
3answers
94 views

If A+tB is nilpotent for n+1 distinct values of t, then A and B are nilpotent.

Suppose A and B are $n\times n$ matrices over $\mathbb{R}$ such that for n+1 distinct $t \in \mathbb{R}$, the matrix A+tB is nilpotent. Prove that A and B are nilpotent. What I've tried so far: ...
0
votes
1answer
18 views

Rank of a simple matrix series

Problem Specifications and Given conditions I have a matrix $L$ with rank 3 and dimension $ 3 \times 3$. $L = K_0+\sum_{n=1}^{\infty}K_i $ . Rank of $K_0$ is 3 and rank of L is also 3. Rank of ...
0
votes
1answer
21 views

Matrix Rank calculation

I have a matrix A . A can be written as A=B+D. I know rank of B. It is 3. Is it possible for A to have ranks $<3$ . If so please prove.
1
vote
1answer
40 views

Rank of a Matrix Sum

I have matrices of $3\times3$ dimension such that S=A+B. I know there is one inequality connecting rank of the matrices A,B and its sum S? Could you write down that here. It will be a great help for ...
1
vote
2answers
50 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
1
vote
2answers
107 views

How to find exponential of triangular matrix

I'm studying for an exam and I can't find this in my notes or in the book, but it's on a past exam... Given $A = \begin{bmatrix}-1 & 1\\0 & -1\end{bmatrix}$, $e^{tA} = \begin{bmatrix}e^{-t} ...
2
votes
0answers
44 views

What do you call the following operations on a symmetric matrix?

Suppose we have a symmetric matrix of the following form, where the diagonal is always zero: \begin{array}{cccc} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0\\ 0 ...
0
votes
0answers
37 views

How to further simplify this equation?

Given that V is an invertible $n$x$n$ matrix and $\Sigma$ is a diagonal rectangular $m$x$n$ matrix, U is an $m$x$m$ matrix, b is an $m$x1 matrix and $\lambda$ is a positive number, how do u further ...
3
votes
2answers
62 views

Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
3
votes
1answer
139 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
0
votes
1answer
33 views

What is the minimum number of sign patterns in $\frac n2$ of columns (or rows) of Hadamard matrices?

Given a Hadamard matrix of size $n$, I want to know what is the minimum number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same. My ...