Articles of jordan normal form

Primary Rational Canonical form Matrix

As I just posted before on Lost on rational and Jordan forms, I’m practising with jordan forms, rational canonical forms… Well, now I’m stuck in a problem because I don’t know where I’m wrong. I have this matrix: $$M=\begin{bmatrix} 4 & -1 & 1 & -1\\ 1 & 3 & 0 & 0\\ 0 & […]

Finding Jordan Canonical form for 3×3 matrix

I was looking at http://www.math.hkbu.edu.hk/~zeng/Teaching/math3407/Jordan_Form.pdf (section 2) $A =\left(\begin{array}{ccc}4 & 0 & 1 \\2 & 3 & 2 \\1 & 0 & 4\end{array}\right)$ We find the Jordan Canonical Basis for $R^3 = \left \{\left(\begin{array}{c}1 \\2 \\1\end{array}\right), \left(\begin{array}{c}0 \\1 \\0\end{array}\right),\left(\begin{array}{c}-1 \\0 \\1\end{array}\right) \right\}$. So far, so good. But I don’t see how they then arrive that […]

Size of Jordan block

Imagine that I’m writing the Jordan form of a matrix and I know that the eigenvalue needs to appear 4 times in the diagonal (algebraic multiplicity is 4) and we need 2 Jordan blocks (geometric multiplicity is 2). Now how do I know the size of the blocks? It could be a 1×1 and 3×3 […]

If we know the eigenvalues of a matrix $A$, and the minimal polynom $m_t(a)$, how do we find the Jordan form of $A$?

We have just learned the Jordan Form of a matrix, and I have to admit that I did not understand the algorithm. Given $A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 2 & 1 & -1 \\ 1 & -1 & 2 & -1 \\ 1 & -1 & 0 […]

Finding Jordan basis of a matrix ($3\times3$ example)

Our teacher didn’t explain us how to find it so I’ve had to look up a bit by myself. I have this matrix : $$A = \begin{pmatrix} 9 & 4 & 5 \\ -4 & 0 & -3 \\ -6 & -4 & -2 \end{pmatrix}$$ I’ve found its characteristic polynomial $p_A(\lambda) = -(\lambda -3)(\lambda – […]

The index of nilpotency of a nilpotent matrix

Let $A$ a matrix in $\mathcal{M}_5(\mathbb C)$ such that $A^5=0$ and $\mathrm{rank}(A^2)=2$, how prove that $A$ is nilpotent with index of nilpotency $4$? Thanks in advance.

Jordan normal form and invertible matrix of generalized eigenvectors proof

Struggling to find a place to start with this proof- just began learning about Jordan normal. Given a 2-by-2 matrix $A$ and a Jordan normal form matrix $J_{\lambda}$, there exists a matrix $S = [v1, v2]$ s.t. $$S^{-1}AS = J_{\lambda}$$ That is, $J_{\lambda}$ is similar to $A$. WTS: $$(A – \lambda I)v_{1} = 0 $$ […]

Jordan canonical form of an upper triangular matrix

Find the Jordan canonical form of the matrix. Justify your answer. $A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 4 \end{bmatrix} $ My Try: The eigenvalues are $1$ and $4$. We have to find $P$ such that $P^{-1}AP=J$. But it is difficult to find two linearly […]

If N is elementary nilpotent matrix, show that N Transpose is similar to N

If $N$ is a $k \times k$ elementary nilpotent matrix, i.e. $N^k = 0$ but $N^{k-1} \ne 0$, then show that $N^\top$ is similar to $N$. Now use the Jordan form to prove that every complex $n \times n$ matrix is similar to its transpose. I have figured out the second part, and am struggling […]

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 those concepts. I’ve been given the following matrix which is associated to an endomorphism $\phi$ of $V=\mathbb{R}^{4}$ $$M=\begin{bmatrix} 1 & 0 & […]