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), ...
0
votes
0answers
10 views
Geometric effect of homogeneous transformation
Describe the geometric effect of the following homogeneous transformation matrix:
$$\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 ...
0
votes
2answers
16 views
Quick question about proofs of theorem concerning Jordan basis
I have a question about proofs of this theorem:
Let $K$ be an algebraically closed field, $V$ be
a finite-dimensional space over $K$ and $f : V → V$ be a linear operator.
Then there exists a Jordan ...
3
votes
3answers
32 views
Matrix being not diagonalizable in F2
We were talking about how the symmetric matrix
$A=\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
is not diagonalizable in the field consisting of only 0 and 1, since the eigenvalues are 0 and ...
1
vote
1answer
55 views
Condition number of a $9\times9$ matrix
would like someone to look over this and assure me I'm not making a silly mistake....
Given a $3\times9$ matrix $V$:
$$
\small\begin{bmatrix}
1.0814 & -0.1251 & -0.1726 & -1.4443 & ...
1
vote
1answer
24 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
27 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
41 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
18 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
15 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
29 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
25 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
37 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
56 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?
0
votes
1answer
67 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?
