All Questions
Tagged with matrix-decomposition jordan-normal-form
54
questions
1
vote
0
answers
35
views
Decomposing a matrix $M$ in the form $M = P^{-1}QP$ where $Q$ and $P$ are real matrices and Q is as diagonal as possible
I am currently working on a tiny matrix library in C++ to help myself learn more about them. So far, I have implemented basic functions such as addition, subtraction, multiplication, the determinant, ...
- 19
1
vote
0
answers
44
views
Is there an example that a specific matrix has Jordan block $J_{2}(i)$?
Consider a matrix of the form $\begin{bmatrix}
A & C\\
-C^{T} & B
\end{bmatrix}$ where A and B are symmetric matrices.
Can matrices of this type have a Jordan normal form representation ...
- 11
3
votes
1
answer
137
views
Order of eigenvectors within basis for Jordan Normal Form?
I'm currently baffled as I thought that the order of eigenvectors within the basis of a JNF decomposition doesn't matter. I may have a made a mistake in my working, but if not, is there a general rule ...
- 33
1
vote
1
answer
67
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Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (https://doi.org/10.48550/arXiv.2006.16364), which seemed to be well-founded. For commuting matrices $\mathbf{A}$ ...
- 528
0
votes
0
answers
45
views
Proof of formula for functions of matrices using Jordan cannonical form
I am having trouble comprehending the proof for Theorem 11.1.1 (p.557) in Matrix Computations by Golub and Van Loan. The theorem is that after "reducing" a square matrix $A$ into Jordan ...
0
votes
1
answer
94
views
Terminology and Structure of Jordan Normal Form
In all of the written resources I have looked at regarding Jordan normal forms of matrices, the Jordan normal form $J$ is defined as having a block structure
$$
J =
\begin{bmatrix}
J_1 & & \\...
- 131
1
vote
2
answers
291
views
Advanced Book on Linear Algebra
After I took a Linear Algebra class I often found many Linear Algebra results that weren't covered in the class. I would like to learn these results therefore I am looking for a book, or even Notes ...
- 1,374
1
vote
0
answers
61
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Why is Jordan normal form possible?
We know that we are able to put a 2x2 matrix $A$ into the following Jordan normal form: $A=PJP^{-1}$ with
$J = \begin{pmatrix}
\lambda_1 & a \\
0 & \lambda_2
\end{pmatrix}$
Where $a=0$ or $a=1$...
- 3,197
1
vote
1
answer
188
views
Find the Jordan canonical form and an invertible $Q$ such that $A=QJQ^{-1}$
$$ A = \begin{bmatrix}
-3 & 3 & -2 \\
-7 & 6 & -3 \\
1 & -1 & 2
\end{bmatrix} $$
The characteristic polynomial can be found to be $p(t)= -(t-1)(t-2)^2$. For
$t=1$, I have that ...
- 1,578
1
vote
2
answers
271
views
Square roots of the basic Jordan block of order $n$ associated with the eigenvalue $1$
Let $F$ be a field. The basic Jordan block of order $n$ associated with the eigenvalue $1$ will be denoted by $J_n$ where
$$J_n=\begin{pmatrix}1&1&&\\&1&1\\ &&\ddots&\...
- 727
1
vote
2
answers
1k
views
Simple proof of Jordan normal form
A lot of proofs in linear algebra use the fact that any square matrix can be written in Jordan normal form.
Unfortunately I can't see why this is the case, I didn't get what Wikipedia said and I just ...
- 147
2
votes
0
answers
69
views
Finding generalized eigenvectors of a matrix
I would like to know how to find the generalized eigenvectors to the following matrix $A$, so that I can express $A$ as $PJP^{-1}$. $$ A = \begin{bmatrix} 1 & -3 & 1\\ 1 & 5 & -1\\2 &...
- 21
1
vote
1
answer
54
views
Size of Jordan blocks according to the characteristic polynomial
Consider a Jordan matrix $\phi$ with the characteristic polynomial
$$\chi_\phi(t) = \prod_{i=1}^m(t - \lambda_i)^{n_i}$$
where $\lambda_i \ne \lambda_j$ for $i \ne j$. I want to show that $n_i$ is the ...
- 147
0
votes
1
answer
98
views
Determining the Jordan decomposition of a given matrix
$
\newcommand{\m}[1]{\left( \begin{matrix} #1 \end{matrix} \right)}
\newcommand{\l}{\lambda}
$
I have the matrix given as follows:
$$A := \m{ -2 & -1 & 1 & 2 \\ 1 & -4 & 1 & 2 \...
- 45.9k
0
votes
0
answers
40
views
Can I substitute letters for long expressions in 5x5 matrix for Jordan decomposition?
I have a 5x5 matrix with long expressions, containing 15 variables. In Mathematica, taking Jordan decomposition of the original matrix makes no progress after one day. If I substitute 25 letters A-Z ...
- 3