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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5 views

Frobenius norm minimization problem

Assume we have a matrix H and SVD has been applied to it and it is of the form H=UXV'. Also, there is a optimum matrix F_opt which consists left eigen values of V i.e. it is a function of V. How can ...
1
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1answer
17 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, ...
2
votes
1answer
11 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 ...
0
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0answers
12 views

Integration of a matrix by MATLAB

How do I integrate a matrix in MATLAB: A=[1,2;3,4]; B=[2*t;t^2]; integral{expm(A*s)*B(s)}ds between the interval [1,t] t can be a symbolic parameter. Thank you so much.
0
votes
3answers
550 views

Every positive definite matrix can be written as $B^TB$ for some invertible $B$

Let $A$ be a positive definite symmetrix matrix. Show that there exists an invertible matrix $B$ such that $A=B^TB$. [Hint: Use the Specral Theorem to write $A = QDQ^T$. Then show that D can be ...
6
votes
1answer
74 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. Can anybody give ...
11
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2answers
457 views

How many $n\times m$ binary matrices are there, up to row and column permutations?

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations. If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...
0
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2answers
34 views

Matrix of Functions

I was thinking about a problem and I realized for a family of functions say $L^1([a,b])$ one can define matrices with functions as elements: $$ A=\left[ \begin{array}{ccc} f_{11}& ...
0
votes
1answer
26 views

Lipschitz continuity of inverse

Given a function f : $\mathbb{R}^n\to\mathbb{R}^m$, which is known to be Lipschitz continuous, can we say anything about the Lipschitz continuity of it's inverse function (in this case, the ...
0
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0answers
11 views

Implementing the Delta method to assess the confidence and prediction intervals

I want to calculate the table of confidence and prediction intervals for a custom Cumulative Distribution Function or CDF, and I am following the forums and articles aid. My major cuestions that I ...
-2
votes
0answers
16 views

Stability of a block matrix with a stable upper left corner

Given a $n\times n$ matrix $A$ is stable, can it be proven that $G=\left(\begin{array}{cc}A & B\\C & -d\end{array}\right)$, where ...
1
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0answers
47 views

Projective special linear groups

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
3
votes
1answer
130 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 ...
-3
votes
0answers
12 views

Maximum absolute column sum of the matrix norm inequality [on hold]

$A=(a_{ij})\in \mathbb{C}^{n\times n}$, $\upsilon(A)=n \max_{ij} |a_{ij}|$ is matrix norm. Prove that $\|A\|_{1} \leq \upsilon(A) \leq n \|A\|_1$.
4
votes
2answers
144 views

Diagonalization of $M=ab^t+ba^t$

Given $a=(a_i)_{i=1}^n$ and $b=(b_i)_{i=1}^n$ in $\mathbb{R}^n$, we define the matrix $M=(m_{ij})_{i,j=1}^n$ as: $$ m_{ij}=a_ib_j + a_jb_i, $$ or equivalently $$M=ab^t+ba^t.$$ What are the eigenvalues ...
5
votes
4answers
329 views

If $A^2=2A$, then $A$ is diagonalizable.

My brain has already burned. I think, I should use a double linear transformation but can't find any proper solution. Let $\mathbb F$ be a field and $A\in M_n (\mathbb F)$ satisfying the equation $$ ...
12
votes
2answers
244 views

$AB=BA$ implies $AB^T=B^TA$.

I am looking for an elementary proof (if such exists) of the following: $$ AB=BA \quad\Longrightarrow\quad AB^T=B^TA, $$ where $A$ and $B$ are $n\times n$ real matrices, and $A$ is a normal matrix, ...
16
votes
3answers
454 views

Prove that $\det(A+B)=\det B$

Assume that the matrices $A,\: B\in \mathbb{R}^{n\times n}$ satisfy $$ A^k=0,\,\, \text{for some $\,k\in \mathbb{Z^+}$}\quad\text{and}\quad AB=BA. $$ Prove that $$\det(A+B)=\det B.$$
2
votes
0answers
29 views

Counting symmetric binary matrices with constant line-sum

I'm trying to count, as the title suggests, symmetric matrices with entries $0$ or $1$ and constant line-sum $k$. ($0 \leq k \leq n$). If you start listing the number of these on a table you'll get a ...
6
votes
1answer
286 views

Cones of positive semidefinite matrices generated by matrices of rank $1$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive ...
2
votes
3answers
95 views

Solving a system of three linear equations with three unknowns

Is my working correct or am I completely wrong? Have I missed anything out? Any feedback is appreciated. Question: Consider the following system of equations $2x + 2y + z = 2$ $−x + 2y − z = −5$ ...
1
vote
0answers
316 views

Solving large, sparse system of linear equations

I have a system of linear equations as follows: $$(A+I)x=B$$ where $I$ is the $n\times n$ identity matrix, $A$ is a $n\times n$ matrix such that the first and last rows are blank, and, for every ...
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0answers
44 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
votes
1answer
20 views

If two covariance matrices commute, is their product a covariance matrix?

Let $A$ and $B$ be two covariance matrices such that $AB=BA$. Is $AB$ a covariance matrix? A covariance matrix must be symmetric and positive semi definite. The symmetry of $AB$ can be proved as ...
1
vote
1answer
85 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 ...
1
vote
1answer
59 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
0answers
7 views

How to write a function to input two matrices to use them in a generic program in SCILAB? [on hold]

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
votes
2answers
53 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 ...
1
vote
2answers
70 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, ...
3
votes
1answer
62 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
2answers
16 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 - ...
0
votes
0answers
32 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
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0answers
20 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 ?
2
votes
2answers
277 views

Find the matrix A of the linear transformation T(M)

I know that if I substitute the first matrix for $T(M)$ I see what T does to each of the basis vectors. I don't understand how that creates a $3\times 3$ matrix though. I was looking at this ...
0
votes
3answers
70 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 ...
6
votes
2answers
95 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 ...
0
votes
1answer
13 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 ...
0
votes
1answer
73 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 $? ...
2
votes
1answer
361 views

Reflection Matrix linear algebra

I am practicing some linear algebra question to prepare for my test. I have come across one question that has given me much trouble. It states: If $\lVert u\rVert = 1$, then $Q = I - 2uu^T$ is a ...
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 ...
8
votes
1answer
213 views

Möbius function from random number sequence

Consider some arbitrary number sequence like the decimal expansion of $\pi$ = {3, 1, 4, 1, 5, 9, 2}. Prepend the sequence with the number $1$ so that you get {1, 3, 1, 4, 1, 5, 9, 2}. Then plug it ...
6
votes
3answers
234 views

Properties of trace $0$ matrices: similarity, invertibility, relation to commutators

$1.$ Are trace $0$ matrices always of the form $AB-BA$? $2.$ Is a trace $0$ matrix over the complex field always similar to a matrix with $0$ as a diagonal element? $3.$ Is a trace $0$ matrices over ...
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
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0answers
20 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
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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 ...
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votes
5answers
120 views

Definition of determinant [closed]

Determinant is a certain function from the set of all $n\times n$ matrices to the set of scalars. How is the determinant defined? What characterizes the determinant function?
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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
574 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
63 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 ...
1
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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 ...