# Questions tagged [jordan-normal-form]

This tag is for questions relating to the Jordan normal form, also known as a Jordan canonical form or JCF of a matrix which is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

1,329 questions
Filter by
Sorted by
Tagged with
1 vote
47 views

### Is the real Jordan form unique?

Let $A\in M_n(\mathbb{R})$. I know that $A$ has a real Jordan form which is obtained from the complex Jordan form by using the usual Jordan blocks for the real eigenvalues and by associating to the ...
• 121
37 views

### How can I find possible non-symmetric $A$ if $A^k$ is symmetric?

Assume $\bf A\in \mathbb R^{n\times n}$ If I know ${\bf A}^k$ and that it is symmetric, how can I systematically find the $\bf A$ which are not? Own work One approach I have considered is to assume a ...
• 25.7k
23 views

• 1,070
38 views

### Confusion about the primary decomposition theorem in linear algebra

I am currently studying the Jordan canonical form which uses the primary decomposition. I have seen the generalised eigenspace decomposition and I know that the algebraic multiplicity which appears in ...
1 vote
50 views

### given the minimal polynomial of matrix $A^2$ find minimal polynomials and Jordan forms of $A$

I'm refreshing some linear algebra and came across this question: Consider a matrix $A \in \mathbb{C}^{4\times 4}$ and suppose the minimal polynomial of its square is $\phi_{A^2}=(x-1)^2$. Now we're ...
• 85
1 vote
65 views

### finding Jordan canonform of $A^3$ given the minimal polynomial of $A$

I'm having trouble with the following question: we were given the following minimal polynomial of a matrix $A\in \mathbb {C}^{4 \times 4}: X^4-2X^3+X^2$. Now we are asked to find the Jordan canonical ...
• 85
1 vote
33 views

### Effect of matrix multiplication on ordering of absolute angle between vectors

Consider two vectors $\mathbf u,\mathbf v$. Let $\theta_0$ be the angle between them. Now multiply them by some matrix $M$. Let $\theta_1$ be the angle between $M\mathbf u$ and $M\mathbf v$. W.L.O.G ...
• 6,248
33 views

### Relation between nilpotent matrix and matrix with simple elementary divisors

I am studying the paper, Goto, Morikuni, On algebraic Lie algebras, J. Math. Soc. Japan 1, 29-45 (1948). ZBL0038.02104. Here the author proves the unique decomposition of a matrix into a nilpotent ...
62 views

### What exactly is the Jordan Canonical Form for the following matrix?

Here is the matrix I am trying to find its JCF: $$\begin{pmatrix} 1 & 2 & 0 & 0\\ 0 & 1 & 2 & 0\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 1 \end{pmatrix}$$ I ...
• 999
24 views

### Deduce Jordan Normal Form from PID finitely generated module structure.

Let $k$ be an algebraically closed field. Any $k$-linear map $\phi:k^n\to k^n$ imposes an extra $k[x]$-module structure on $k^n$ by defining $p(x)\cdot v = p(\phi)(v).$ Clearly then $k^n$ is a ...
• 413
29 views

### Show supremum norm of solution to ode is less than $k$ i.e. $||e^{At}|| \leq k$.

So I'm trying to make an exercise in ode's. Let $A \in M_5(\mathbb{R})$ be a matrix with 3 eigen-values $\lambda_1 = -1$ with multiplicity $3$ and $\dim(\text{Ker}(A+I))=1$, $\lambda_{2,3} = \pm 2i$. ...
14 views

### All possible block diagonal forms for real valued $4\times 4$ matrices

This is one part of a larger question I'm trying to answer. The first part was determining all the possible Jordan normal forms for complex $4\times 4$ matrices, and the second part was determining ...
• 69
10 views

### Proof for algorithm of Jordan normal form using cyclic basis

Suppose $A\in M_n(\mathbb R)$ with \begin{align} \big|A-\lambda I_n\big|\,&=\,\pm(\lambda-\lambda_1)^{m_1}\cdots(\lambda-\lambda_p)^{m_p} \newline minpoly_A(\lambda)\,&=\,(\lambda-\lambda_1)^{...
• 409
1 vote
139 views

### Solve the matrix equation for $X$

Solve $X^{7}=\begin{pmatrix} 1 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$. We dont know the field where the entries of $X$ are. The matrix $A=X^{7}$ is not ...
• 1,251
38 views

### Numerical inaccuracy in diagonalizing a matrix / finding its Jordan form

I was trying to numerically compute eigenvalues and eigenvectors of a matrix in Python with Numpy. The problem was that the inverse similarity transformation of the eigenvalue matrix does not ...
44 views

### Finding all possible Jordan forms from the Characteristic polynomial

Let A be a 7 x 7 matrix with characteristic polynomial $(t − 2)^4(3 − t)^3$. It is known that in the Jordan form of A, the largest blocks for both the eigenvalues are of order 2. Show that there are ...
• 21
43 views

### Is there some geometric understanding of a Jordan normal matrix?

We know that an eigenvalue decomposition of a matrix is to find those eigenvectors that are just scale for some coefficients. But what about Jordan matrix decomposition? I just learn how it is ...
• 31
38 views

### Question on a simple proof of Jordan Normal Form

I am currently a year 2 uni mathematician, understanding the Jordan Normal Form in Linear Algebra. Here is a paper for a short proof of the Jordan Normal Form, which I am looking at in order to ...
• 423
74 views

### Suppose $AB=BA$ and $x$ is an eigenvector of $A$ with geometric multiplicity $k$, can $Span\{x, Bx, ...\}$ be represented?

Suppose $A$ and $B$ are $n \times n$ commuting matrix: $AB=BA$. $x$ is an eigenvector of $A$ with the eigenvalue $\lambda$ of geometric multiplicity $k$ i.e. $\ker(\lambda I-A)$ is of dimension $k$. ...
• 159
24 views

### Possible non similar jordan Forms

Be A an M_6(R) matrix so that A^4-8A^2 +16I = 0. What are the possible Jordan Forms non similar to A? I tried to solve that, so I arrived in the equation (A-2I)^2*(A+2I)^2 = 0. So this will give me ...
• 13
45 views

• 323
1 vote
99 views

### Commuting matrices and their Jordan forms

I am recently studying commuting matrices. I was reading the book Invariant Subspaces of Matrices with Applications(Godberg, Lancaster and Rodman) and pg.295-296 claims that a matrix $X$ is a solution ...
31 views

### Proving that every invariant subspace of $\mathbb{F}^n$ is of the form $span(e_1,...,e_k)$

Let $T_{J_n(0)}:\mathbb{F}^n \to \mathbb{F}^n$. I need to prove that every invariant subspace $W$ of $\mathbb{F}^n$ is of the form $Span(e_1,...,e_k)$ for some $0\leq k \leq n$. This is what I've got ...
• 323
$\frac{d}{d Y}: K[Y] \rightarrow K[Y], \quad \sum_{i \geq 0} a_{i} Y^{i} \mapsto \sum_{i \geq 0}(i+1) a_{i+1} Y^{i}$ $P_n =\[f \in K[Y] \mid \operatorname{deg}(f) \leq n]$ (this is a set) \$f=\left. \...