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), ...
1
vote
1answer
44 views
How can one define the trace of a linear operator on any finite dimensional vector space, using the fact that $tr(A) = tr(P^{-1}AP)$?
Firstly, I had to prove that $tr(AB) = tr(BA)$ and deduce that the trace is an invariant of similarity i.e. that $tr(A) = tr(P^{-1}AP)$ for any $A$ and invertible $P$. I could prove the first part - ...
1
vote
0answers
13 views
Convert hermitian matrix to symmetric
Is there some simple transformation (or a simple way to find it) which would convert any given hermitian matrix $A$ to a symmetric matrix $B$ with the same spectrum as that of $A$ (so I guess that ...
0
votes
0answers
24 views
What is the error in Newton's Method for Matrix Inversion?
I need it to invert a matrix. Wikipedia explains that there is a generalization of the Newton Method for matrices. However, there is nothing mentioned about the error bounds.
Suppose we have, as ...
4
votes
0answers
48 views
Invariant of matrix under elementary transformations
$\DeclareMathOperator{\rank}{rank}$
Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix
$$
B = \begin{bmatrix} A & b \\ b^T & c ...
0
votes
0answers
14 views
when gives svd decomposition matrices,whose element are different from 0 and 1
i want to understand one basic question,what is a criteria that for given matrix,whose element are just only $1$ and $0$,it's svd decomposition gives me matrices, whose element are different from ...
1
vote
2answers
36 views
Linear Transformations: Scaling along the line $y=x$
What is the geometric meaning of scaling an object by a factor $k$ along the line $y=x$?
What will be the shape of a square with vertices $(2,1)$, $(3,2)$, $(3,1)$, and $(2,2)$ if it is scaled by a ...
1
vote
1answer
45 views
Cancellation property in matrices.
I just found a question which is based on a doubt I have carrying for over 10 years.
If $ACC^t=BCC^t$ : $C^t$ means transpose of $C$
Is
$A=B$
$AC=BC$
Sorry if this is a trivial question.
...
1
vote
0answers
36 views
When is a matrix called well-ordered?
We have a quick question, looking for information and/or references links.
Is there a more specific mathematical definition/criteria for when matrices can be called well ordered or totally ordered? ...
0
votes
1answer
31 views
Name of type of Matrix
I am struggling to remember the name of this type of Matrix.
Could anyone assist?
$$
\begin{bmatrix}
0 & 0 & 0 & 1 & 1\\
0 & 0 & 1 & 0 & 1\\
0 & 1 & 0 & 0 ...
6
votes
5answers
159 views
matrices $A$ of order $2 \times 2$ that satisfy the equation $A^2-5A+6I = O$
Find all matrices $A$ of order $2 \times 2$ that satisfy the equation $A^2-5A+6I = O$
My try:: Given $A^2-5A+6I = O\Leftrightarrow A^2-3A-2A+6I^2 =O$
So $A=3I = \begin{pmatrix}
3 & 0\\
0 & ...
0
votes
0answers
15 views
How will an orthogonal array look for 3 levels and 3 factors?
I understand that an orthogonal array with 3 factors (parameters) will have 3 columns and if there are 3 levels, then it means each parameter can have 3 values.
However, when using a selector from ...
0
votes
1answer
59 views
Prove that $(X'X)^{-1}X'AX(X'X)^{-1}-(X'A^{-1}X)^{-1}$ is positive definite
How to prove if A is a positive definite matrix, then $(X'X)^{-1}X'AX(X'X)^{-1}-(X'A^{-1}X)^{-1}$ is also positive definite? Here $X'$ denotes the transpose of $X$. $A$ is square and $X$ is $n\times ...
2
votes
1answer
25 views
On triangular decomposition of square matrix
Let $L\in Gl_n(\mathbb{C})$ and define $A=LL^*$. Let us consider another decomposition such as $A=L_1L_1^*$. What is the relation between $L$ and $L_1$. One obvious relation is $L_1=LU$ where $U$ is ...
1
vote
2answers
84 views
The relationship between eigenvalues of matrices $XY$ and $YX$
If $X \in \mathbb{C}^{m \times n}$ and $Y \in \mathbb{C}^{n \times m}$ ($m \geq n$), how to prove that $\lambda (XY) = \lambda (YX) \cup \underbrace{\left \{ 0, ..., 0 \right \}}_{m-n}$? Here, ...
1
vote
1answer
47 views
Linear algebra, Schur set
Can you guys give me some hints on how I can start this problem? Thanks in advance!
Let $ U(n) \subseteq M_n(\mathbb C) $ be the set (group) of all $ n \times n $ unitary matrices. Let $ T ...
0
votes
1answer
19 views
Expressing a transformation matrix
Let $B=\{v_1,...,v_n\}$ and $C=\{w_1,...,w_n\}$ be bases to $V$. Suppose: $w_i=m_{i1}v_1+...+m_{in}v_n$ for $m_{ij}\in F, 1\le i,j \le n$. $M$ is an invertible matrix whose ($i,j$) member is $m_{ij}$.
...
0
votes
2answers
66 views
A matrix is normal, if and only if?
Let $A \in M_n(\mathbb C)$. Let $\langle \; \cdot\; , \; \cdot\; \rangle$ be the standard inner product in $ \mathbb C^n$, viewed either as row vectors or as column vectors.
Let $r_j$ be the $j$-th ...
3
votes
0answers
47 views
Name of a special matrix
I have a matrix which is kind of symmetrical with the other diagonal, i.e., something like
$$A = \left[ \begin{array}{c c c c}
a & b & c & d \\
e & f & g & c \\
h ...
1
vote
2answers
28 views
Is the linear dependence test also valid for matrices?
I have the set of matrices
$
\begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix}
$
$
\begin{pmatrix}
0 & 1 \\
0 & 0 \\
...
1
vote
1answer
39 views
How would I find this eigenvalue?
I'm told to let $A$ be the matrix of the linear transformation $T$ and without writing $A$, find an eigenvalue of $A$ and describe the eigenspace. The first is to let $T$ be the transformation on ...
1
vote
1answer
75 views
Limit of matrix powers.
Consider an arbitrary matrix $A$ with eigenvalues within the unit circle. Is there a nice formula for $A^\infty = \lim_{n \rightarrow \infty} A^n$?
In particular, maybe there is a formula which ...
1
vote
1answer
53 views
problem with 4x4 matrix with big elements
I have a homework for my linear algebra class at my university the thing is that we get a 4x4 matrix A then we have to find it's Transpose which is pretty easy and then find the matrix B=(A^T)*A also ...
0
votes
0answers
16 views
Geometric effect of homogeneous transformation
Describe the geometric effect of the following homogeneous transformation matrix:
$$\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 ...
0
votes
2answers
26 views
Quick question about proofs of theorem concerning Jordan basis
I have a question about proofs of this theorem:
Let $K$ be an algebraically closed field, $V$ be
a finite-dimensional space over $K$ and $f : V → V$ be a linear operator.
Then there exists a Jordan ...
3
votes
3answers
66 views
Matrix being not diagonalizable in F2
We were talking about how the symmetric matrix
$A=\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
is not diagonalizable in the field consisting of only 0 and 1, since the eigenvalues are 0 and ...
1
vote
1answer
70 views
Condition number of a $9\times9$ matrix
would like someone to look over this and assure me I'm not making a silly mistake....
Given a $3\times9$ matrix $V$:
$$
\small\begin{bmatrix}
1.0814 & -0.1251 & -0.1726 & -1.4443 & ...
1
vote
1answer
53 views
Linear Algebra: Least-Squares Approximation & “Normal Equation”
I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] ...
2
votes
1answer
32 views
I need to diagonalize this matrix but I'm not sure it can be
This is the matrix I need to diagonalize: $A=\left[\begin{matrix}3&2\\0&3\end{matrix}\right]$. So I found the eigenvalue by taking the determinant of $(A-\lambda I)$ and solving for ...
1
vote
1answer
46 views
Find all eigenvalues and corresponding eigenvectors for the matrix?
Find all eigenvalues and corresponding eigenvectors for the matrix
$$
\left(\begin{array}{cr}
0&-1 \\
2&3
\end{array}\right)
$$
Not looking for a answer but i dont know what a "eigenvalues" is ...
1
vote
1answer
43 views
How to combine covariance matrices?
I have a data set of points in three dimensions.
I'm calculating the barycenter (mean) and $3\times3$ covariance matrix from this data set.
I store the average, the $3\times3$ matrix (where really ...
0
votes
0answers
19 views
conditional matrix form
I am deriving a matrix form for part of an equation which demands a conditional form but I have trouble in making it so that to be acceptable in scientific communities. Let's assume that we have ...
2
votes
2answers
34 views
Picard iterations of a matrix
I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one.
We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
1
vote
1answer
38 views
how to calculate the “variance OF the covariance” matrix : E[vech(x x') vech(x x')'] for normal distributed x?
Supposing a vector x follows normal distribution. I want to calculate the expectation of the "variance Of the covariance matrix" (not variance-covariance matrix) in a vector form, meaning E[vech(x ...
2
votes
0answers
54 views
Why matrix representation of convolution cannot explain the convolution theorem?
A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
0
votes
2answers
50 views
Matrix Algebra (Elementary)
I have $\hat\xi =\lambda_1\textbf{1V}^{-1} + \lambda_2\textbf{rV}^{-1}$ and sub it in to my two constraints, namely, $\xi\textbf{1}^T = 1$ and $\xi\textbf{r}^T = \mu$. My lecture notes then say set ...
0
votes
1answer
63 views
Eigenvector with eigen value of 1
How is an eigenvector with eigen value of 1, say v, multiplied by its transpose the identity matrix? v' * v = I?
1
vote
1answer
120 views
How to prove that $D := ABC$ is also positive definite?
Let $A,$ $B$ and $C$ be symmetric, positive definite matrices and suppose that $D := ABC$ is symmetric.
How might I prove that $D$ is also positive definite?
0
votes
2answers
49 views
Volume of a parallelepiped, given three vectors
I want the volume of a parallelpiped and I have the three vectors $$4e_1+2e_2-e_3$$$$e_1-3e_2-2e_3$$$$2e_1-e_2+3e_3$$ that coinciding with three of the parallelpipeds sides. HON-base
I made it into a ...
0
votes
1answer
43 views
Is the sum of two singular matrix also a singular matrix
If $A$ and $B$ are singular and both $n\times n$, is $A+B$ also singular?
0
votes
1answer
42 views
Proof of adjoint(ab) = adjoint(b)adjoint(a)
So I'm trying to prove whether
adjoint(AB) = adjoint(B)adjoint(A).
My tactic is to look at the ijth entry of both matrices and compare, but the fact that AB is being transposed is confusing me.
...
0
votes
0answers
26 views
Subgradient of matrix $l1$-norm
Let $X$ be a square matrix, what is the sub-gradient of $f(X) = ||AX-XB ||_{l1}$?
$A$ and $B$ are both constant matrix.
I am very confuse about the chain rule on matrix derivatives.
1
vote
2answers
48 views
Identifying matrix vector multiplication
I have the following question in a book:
According to the book, the answer is (D). But I don't understand how. Isn't this just scalar multiplication? The solution in the book says that I have to ...
0
votes
1answer
32 views
finding the P matrix (diagonalization of a matrix)
I'm trying to find the diagonalization of a matrix :
this is my matrix :
$$
A =\begin{pmatrix}
0 & -1 & 0 \\
-1 & 0 & 0 \\
1 & 1 & 1 \\
...
0
votes
1answer
43 views
diagonalization of a matrix
I want to diagonalize this matrix :
$$
A =\begin{pmatrix}
0 & -1 & 0 \\
-1 & 0 & 0 \\
1 & 1 & 1 \\
\end{pmatrix}
$$
the first step I ...
1
vote
1answer
23 views
Definition of minimal and characteristic polynomials
I have defined the characteristic and minimal polynomial as follows, but have been told this is not strictly correct since det$(-I)$ is not necessarily 1, so my formulae don't match for $A=0$, how can ...
1
vote
1answer
63 views
Get code words from generator matrix
I have some issue regarding the generator matrix. Please can some body can explain me "How to get Codebook from Generator matrix?"
Following is my issue
Generator matrix has 3 code words.
Then ...
0
votes
1answer
17 views
determining the associated matrix has an endomorphism of R ^ n.
What is the method for determining the associated matrix has an endomorphism of R ^ n.
for example:
f is defined on i³ with : ∀(x,y,z) ∈ i³ , f(x,y,z) = (x+2z, 2y, 2x+z)
The canonical basis of i³ ...
0
votes
1answer
44 views
Solution of a system of linear equations with n variables
I have a system of linear equations with n variables
\begin{cases}
a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = \frac{1}{2}x_1\\[4pt]
a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = \frac{1}{2}x_2\\[4pt]
...
1
vote
0answers
20 views
Improve Upper Triangular Matrix Computation for Gauss Jordan method?
I am solving Simultaneous Equations using the Gauss Jordan method.
I am having a problem in computing the Upper triangular Matrix with sufficient accuracy for no of variables >50.
Some of the elements ...
10
votes
2answers
61 views
Rank of the difference of matrices [duplicate]
Let $A$ and $B$ be to $n \times n$ matrices. My question is:
Is $\operatorname{rank}(A-B) \geq \operatorname{rank}(A) - \operatorname{rank}(B)$ true in general? Or maybe under certain assumptions?
