Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, Hamel basis, dimension, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization...
1
vote
0answers
11 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 \\
...
0
votes
1answer
8 views
Underdetermined linear systems least squares
I have an underdetermined linear system, with 3 equations and four unknows. I also know an initial guess for these 4 unknows. The article I am reading says: We can solve the system using the least ...
1
vote
1answer
22 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 ...
0
votes
0answers
16 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 ...
3
votes
0answers
21 views
Angular alignment of points on two concentric circles
I have two concentric circles $C_1$ and $C_2$ with radii $r_1,r_2$ such that $r_1< r_2$and a set of finite points $P=\left \{ p_1,p_2...p_n \right \}$ and $Q=\left \{ q_1,q_2...q_n \right \}$ are ...
3
votes
5answers
76 views
Prove that if $A - A^2 = I$ then $A$ has no real eigenvalues
Given:
$$ A \in M_{n\times n}(\mathbb R) \; , \; A - A^2 = I $$
Then we have to prove that $A$ does not have real eigenvalues.
How do we prove such a thing?
1
vote
1answer
10 views
Angle consistency between vectors in N dimensions
I am trying to understand how rotations work in higher dimensions. Let us assume we have a set of points $p_i\in P$ in $N$ dimensions, related to another set of points $q_i \in Q$ by a rotation $R$. ...
2
votes
1answer
37 views
Show AB and BA have the same eigenvalues [duplicate]
If $A$ and $B$ are $n$ by $n$ matrices show that $AB$ and $BA$ have the same eigenvalues. I see why this is true if both are nonsingular. But does it still hold if they are not invertible?
Thanks!
2
votes
2answers
19 views
Finding for every parameter $\lambda$ if matrix is diagonalizable
Given:
$$A = \begin{pmatrix} 1 & i & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & i \end{pmatrix} \; , \; \lambda \in \mathbb C$$
For every value of $\lambda$ I have to know if the matrix ...
0
votes
4answers
58 views
Prove that if $A \in M_{2\times2}\mathbb {(R)}$ is symmetric then A is diagonalizable
Given that: $$A \in M_{2\times2} \mathbb {(R)}$$
we have to prove that $A$ is diagonalizable.
As in:
$$\text{There exists a turnable matrix } P \; (\text{det(P) != 0 }) \; \text{such that}:$$
...
-1
votes
0answers
18 views
How to calculate the lens distortion cordinates with a known coefficients?
I have $k_1$ and $k_2$
$$
r^2=x^2+y^2
\\
x' = (1 + k_1r^2 + k_2r^4)x\\
y' = (1 + k_1r^2 + k_2r^4)y
$$
How to calculate x' and y'?
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 ...
0
votes
0answers
15 views
Inverse of Eigen value
What is the physical meaning of inverse square root of the eigen value? Is it possible to use it as stretch factor to decorrelate the data.
2
votes
2answers
29 views
Minimum distance of the linear code $\{0,1\}$
Let $H$ be a check matrix for a linear code $C$. Then the minimum distance
of $C$ is $d \in \mathbb N$ such that there exists a set of $d$, but no set of $d-1$, linearly dependent columns in $H$.
...
3
votes
0answers
14 views
Isomorphism between $E_8$ lattice and lattice defined by Extended Hamming Code
I have read that the following two lattices are isomorphic, and of course it seems believable, but it would be nice to have a sketch of how to construct the bijection.
Let $C$ be some extended ...
