Questions tagged [jordan-normal-form]

The Jordan normal form of a matrix is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

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Classification of bilinear forms: operator $A^{-1} A^T$ for bilinear form $A$

I would like to understand a classification of non-degenerate (not necessary symmetric or skew-symmetric) bilinear forms over an algebraically closed field via an operator $\kappa=A^{-1} A^T$ for a ...
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Lost on rational and Jordan forms

I'm having a lot of trouble trying to understand rational canonical form, primary rational canonical form, and Jordan form. I've looked at the posts about this, but I haven't been able to understand ...
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Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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Jordan normal form theorem proof question

Theorem: Assume that the characteristic polynomial $x_f$ splits into linear factors. Then there exists a Jordan normal form for f. The Jordan normal form is unique up to the order of the Jordan blocks....
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Given a Jordan canonical basis, how to find out to which generalized eigenspace picked generalized eigenvector belongs

Suppose we have finite-dimensional linear operator $A:V\to V$ , that has eigenvalues $\lambda_1 ,\lambda_2, ... \lambda_n$ . It is known that we can decompose $V$ into direct sum of generalized ...
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Proving that a matrix is nonnegative if its powers are nonnegative

I am working on a problem involving doubly stochastic matrices where I must prove that $P$ is doubly stochastic if and only if $P^k$ is doubly stochastic for $k = 1, 2, ...$ It is easy to show that if ...
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Jordan form of the matrices of a group

Let's consider a set of $m$ generic square matricies $(N;N)$ defined on $R$ which forms a group. Chosen one of these $m$ matrices, I know that, by changing the base on my vectorial space, I can ...
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What is the difference between ones and zeroes on the superdiagonal in a Jordan normal form matrix?

I've observed that in Jordan normal form the eigenvalues are obviously on the diagonal, but on the superdiagonal there can sometimes be only ones, sometimes be only zeroes and sometimes both. What is ...
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Uniqueness of Jordan-Chevalley-Decomposition - why do the nilpotent matrices commute?

The Jordan-Chevalley-Theorem states that for a given endomorphism $f\in \mathrm{End}_K(V)$ of a $K$-vector space $V$, such that the characteristic polynomial $\chi_f$ splits into factors, there exist ...
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Question regarding possibilities for Jordan normal form

In this problem I am given a matrix $$B = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ ...
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A Step in The Proof of Jordan Form: Trivial Space

Let $\mathcal{R}$ denote the range space and $\mathcal{N}$ the null space of a linear transformation. We write $$\mathcal{R}(t^n) = \mathcal{R}(t^{n+1}) = \dots = \mathcal{R}_{\infty}(t)$$ and ...
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Square root of a matrix by looking at the eigenspaces

A problem that is well known in linear algebra is the existence of square root of a matrix, where square root of matrix $A \in M_n(\mathbb F)$ is defined to be $K \in M_n(\mathbb F)$ such that $K^2=A$ ...
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Jordan Normal Form from characteristic and minimal polynomials

I'm trying to find Jordan Normal Form of a linear transformation $F: V \to V$ with characteristic polynomial $$P_{F}(t) =(t+1)^3 (t-1)^3$$ and minimal polynomial $$M_{F}(t) =(t+1)^2 (t-1)^2$$ It has ...
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Proof that any antisymmetric matrix C is congruent to a block diagonal matrix?

Is there a simple proof that shows that, for any antisymmetric $C$, there exists an orthogonal matrix $P$ such that $P^TCP$ is a block diagonal matrix? I have found a couple of very longwinded ...
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Commutativity and Jordan Decomposition

Apologies in advance if my formatting is bad. I'm working on the following problem and I've run into a bit of a wall. Let $V$ be a vector space over an algebraically closed field $\mathcal{F}$....