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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2
votes
2answers
29 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
40 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
13 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
35 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
38 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
38 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
33 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
1answer
32 views

Differentiate function with respect to matrix

I need to differentiate the following function: $$f(\Sigma)=-\frac{1}{2}\log|\Sigma|+-\sum_i C_i\exp\left(-\frac{1}{2}\frac{\mu_i^2}{1+\Sigma_{ii}}\right)$$ to find, $\frac{\partial f}{\partial ...
1
vote
0answers
22 views

Adding matrices with transposition

Let's say I have two arrays as follows: A = [1,2,3] B= [1,2,3] Can I add $A + B^t$ ?
1
vote
3answers
24 views

Can zero rows in matrices be ignored in calculations of matrix products?

I understand that when calculating the product of 2 matrices you need to account for the dimensions. But when there is an empty row in one of the matrices, why does it need to be accounted for? What ...
1
vote
6answers
113 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
0answers
14 views

Is there a bound on largest eigenvalue for covariance matrix of discrete random variable?

I have a random variable $Z=(Z_1,\ldots,Z_p)$. Each component can take values in {-1,0,1}. Is there a way to bound the largest eigenvalue of Cov(Z)? Actually, I have a latent multinormal variable ...
2
votes
2answers
35 views

Solving for a Rotation Matrix Equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$

I would like to solve for an equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$ where $R_1$, $R_2$, and $R_\mathrm{x}$ are 3x3 rotation matrices, $R_1$ and $R_2$ are known, and $R_\mathrm{x}$ is unknown. ...
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 ...
4
votes
1answer
47 views

Geometric interpretation of complex eigenvalues

What is the geometric interpretation of complex eigenvalues? For me it is clear that real eigenvalues of a matrix $A$ are associates to eigenvectors along which the matrix $A$ contracts or expands. ...
1
vote
0answers
13 views

Sparse matrix algorithms involving data-driven or random access / walk

I am looking for some well-known algorithms in which sparse matrix elements are accessed in a non-structured way, i.e. row/column depends on a value of another (sparse) matrix/vector element or some ...
0
votes
1answer
19 views

Convergence of a Cauchy sequence of matrices

I have a Cauchy sequence of matrices $C_i \in R^{p \times q}$, i.e. $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm (I just need the property that $\|C_1-C_2\|>\delta ...
0
votes
0answers
15 views

Ia it possible to use the deflation algorithm to compute the eigenvalues of a large sparse matrix

I am trying to compute the eigenvalues of a large sparse matrix (about 10% of the values are nonzero). The matrix is real valued, but since it is accumulated by a stochastic process it is not fully ...
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 ...
3
votes
2answers
68 views

Matrix notation why is column 3= column 1?

let $A =$\begin{bmatrix}a_{11} & a_{21} & a_{11}\\a_{12} & a_{22} & a_{12}\\a_{13} & a_{23} & a_{13}\end{bmatrix} where $a_{ij}\in\Bbb R$ for each $1\le i , j\le 3$ which of ...
1
vote
2answers
38 views

Derivative of the trace of $X^TP^TPX$ with respect to P

$\newcommand{\Tr}{\operatorname{Tr}}$ Consider the following expression: $\Tr(X^TP^TPX)$ where $X$ and $P$ are real matrices. What is the best way to approach the calculation of its derivative ...
1
vote
0answers
36 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
1answer
24 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]; i.e, how to compute: integral{expm(A*s)*B(s)}ds between ...
0
votes
2answers
42 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
0answers
14 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
18 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 ...
-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 $\|A\|_{1} \leq \upsilon(A) \leq n \|A\|_1$.
1
vote
1answer
60 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 ...
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 ...
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
54 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 ...
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 - ...
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 ?
1
vote
2answers
78 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
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 ...
1
vote
0answers
48 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
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 ...
6
votes
1answer
75 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 ...
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 ...
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
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
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 ...