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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4 views

How to write a function to input two matrices to use them in a generic program in SCILAB?

I am trying to find out who to call for input in the form of matrices...how to set a prompt for entering values of two matrices and then store the values? Thank You.
2
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2answers
47 views

A question about invertible matrices

A square matrix $A$ over the reals is said to be invertible in practice if there exists a matrix $B$ of the same size s. t. all the entries of $AB$ differ from the corresponding entries of the ...
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2answers
13 views

Transformation matrix of a polynomial

I would really appretiate some help about the following transformation matrices. We have to write a tranformation matrix in basis $B = \{ 1 + x, x + x^2, x^2 \}$ with a polynomial $(Ap)(x) = (x^2 - ...
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0answers
17 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 ?
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2answers
62 views

Derivative of matrix product: is it true that $\frac{d}{dt}(A^TA) = 2A^T \frac{dA}{dt}$?

$A$ is a square matrix. All elements of $A$ depend on a parameter $t$, that is, $a_{ij}=a_{ij}(t)$. Let $S(A):=A^TA$, and take the derivative of $S$ w.r.t. $t$: $\displaystyle \frac{dS}{dt}$ Now, ...
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1answer
53 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
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0answers
23 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 ...
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1answer
53 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 ...
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0answers
24 views

Equivalence of Hadamard matrix

This question is from The Theory of Error-Correcting Codes by MacWilliams and Sloane, Problem 2.(3). If n = $2^m$, let $u_1$, $u_2$,...,$u_n$ denote the distinct $m$-tuples. Show that the matrix $H = ...
0
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2answers
48 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 ...
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1answer
61 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 $? ...
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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 ...
4
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1answer
45 views

Searching two matrix A and B, such that exp(A+B)=exp(A)exp(B) but AB is not equal to BA.

We know that if two matrix A and B commutes then exp(A+B)=exp(A)exp(B). I am trying to find two matrix that does not commute but exp(A+B)=exp(A)exp(B) is true for them. my tries say that order of ...
0
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1answer
18 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 ...
6
votes
2answers
90 views

Efficient way to compute $(A+D)^{-1}$ when $A^{-1}$ is known

I need to compute the inverse of a matrix sum $A+D$, where the inverse of $A\in\mathbb{R}^{n\times n}$ is known. The matrix $D\in\mathbb{R}^{n\times n}$ is a diagonal matrix which can be thought of as ...
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3answers
63 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 ...
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0answers
21 views

Gaussian Elimination and Matrix [on hold]

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. \begin{cases} 4x - y + 3z = 12 \\ x + 4y + 6z = -32 \\ 5x + 3y + 9z = 20 ...
3
votes
5answers
559 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 $A,B \text{ have the same inverse matrix} \iff A=B$?
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0answers
10 views

Expanding a product of matrices with tensor product and transpose

I'm trying to expand the following product of $\pm1$ matrices $H_1, H_2, K_1, K_2$: $(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 (H_1-H_2)\otimes K_2^T)(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 ...
3
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1answer
62 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 ...
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votes
0answers
59 views

Series expansion - Involving matrix exponent [on hold]

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 = ...
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1answer
47 views

Matrix exponential of a skew symmetric matrix

I have a skew symmetric matrix $$C=\left( \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \\ \end{array} \right).$$. Question 1) How do I compute ...
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1answer
27 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} ...
4
votes
0answers
58 views

Expectation of the absolut value of the determinant of a random matrix

Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable $$X := |\det A|$$ Can ...
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1answer
79 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 ...
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1answer
11 views

Proximal operator fixed point property for matrices

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ $\newcommand{\dom}{\operatorname{dom}}$ Recall again that the proximal operator for vectors $\prox_{f}: R^n ...
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2answers
69 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
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0answers
24 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 ...
1
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1answer
34 views

How to characterize elements in the Bruhat open cell?

This might be an elementary question. For simplicity, let's assume $G=GL(n,F)$, where $F$ is a local field. Let $U$ be the subgroup of upper triangular unipotents, $A$ the subgroup of diagonal ...
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0answers
19 views

Finding dynamic range of rotation matrices

How do I theoretically calculate the maximum value of the transformed output can reach after transforming a vector? If it is an eigen vector then the eigen value will tell the max scaling possible, ...
0
votes
1answer
28 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: ...
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1answer
22 views

Rotate $xyz$ by use of pitch and yaw around origin

I have a project for a game which uses pitch/yaw for the direction of a players head. The pitch ranges from $0$ to $180$ and the yaw is $0$ to $360$. Yaw modifies $X$ and $Z$, pitch modifies the $Y$, ...
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
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0answers
36 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 ...
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2answers
84 views

Unsolvable(?) Assignment Problem

I've recently been trying to implement the Hungarian Method in C++, and I've been using 5x5 matrices to test my program. Last night I came across a matrix which neither I nor my program can solve. Is ...
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2answers
22 views

Matricial differentiation $x x^{\top} b $

What is the drivative of $x x^{\top} b $ with respect to x, knowing that b is constant vector?
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0answers
7 views

How to perform FFT-SPA decoding?

i am working with Fast Fourier Transforms based Sum Product Algorithm. Actually, I have to code it in MATLAB and Without using MATLAB library FUNCTIONS.. I have used the algorithm from Non Binary ...
4
votes
3answers
91 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: ...
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0answers
17 views

Matrix Theory: Orthogonal and Linear independent [on hold]

Suppose that we want to find a vector c orthogonal to both of a and b (a) Express the condition using two dot products. (b) Find any one vector c other than the zero vector with this property. (c) ...
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1answer
17 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 ...
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1answer
20 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.
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1answer
37 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 ...
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2answers
49 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) ...
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2answers
106 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} ...
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0answers
43 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 ...
5
votes
1answer
50 views

Is S a group under matrix addition

Another matrix question! Let $$S=\{A \in M_2(\mathbb{R}):f(A)=0\}\text{ and }f\left(\begin{bmatrix}a&b\\c&d \end{bmatrix}\right)=b$$ Is S a group under matrix addition. Either prove that ...
0
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1answer
23 views

How to judge if a symmetric matrix can be factorized into two vectors?

How can we judge if this matrix can be written as the product of a column and a row vector? $A=\begin{bmatrix} \alpha & a & \alpha\\ \beta & b & \beta\\ \alpha & a & ...