M y question is relating to the matrix as A. I have started off this problem by finding the eigenvalues, which turns out to be 3 ( I should note that it has an algebraic multiplicty of 3) From there I have found the corresponding Eigenspace which is $E_3=span(-1,1,1)$ I am little confused as to […]

I’m trying to prove: If $J$ is a single Jordan block corresponding to an eigenvalue $\lambda = 1$, then $J^k$ is similar to $J$, where $k$ is a nonzero integer. Moreover, if $\lambda = 1$ is the only eigenvalue of a matrix $A$, then $A^k$ is similar to $A$. Thanks

Let $A = \begin{bmatrix}1&-3&0&3\\-2&-6&0&13\\0&-3&1&3\\-1&-4&0&8\end{bmatrix}$, Prove that $A$ is similar to $A^n$ for each $n>0$. I found that the characteristic polynomial of $A$ is $(t-1)^4$, and the minimal polynomial is $(t-1)^3$. And the Jordan form of $A$ is \begin{bmatrix}1&1&0&0\\0&1&1&0\\0&0&1&0\\0&0&0&1\end{bmatrix} I guess the key to solve this is to use the fact that two matrices are similar […]

I have the matrix $A=\begin{pmatrix}2&2&1\\-1&0&1\\4&1&-1\end{pmatrix}$, I want to write it in Jordan-Normal Form. I have $x_1=3,x_2=x_3=-1$ and calculated eigenvectors $v_1=\begin{pmatrix}1\\0\\1\end{pmatrix},v_2=\begin{pmatrix}1\\-4\\5\end{pmatrix},v_3=\begin{pmatrix}0\\0\\0\end{pmatrix}$. But, the matrix $Z=\begin{pmatrix}1&1&0\\0&-4&0\\1&5&0\end{pmatrix}$ is not invertible since $\text{det}(Z)=0$. Does this mean the matrix cannot be written in JNF or do I need to find different eigenvectors? I have tried to find different eigenvectors, but […]

This question already has an answer here: Are two matrices similar iff they have the same Jordan Canonical form? 2 answers

For instance, with $T \in \mathcal{L}$(Mat($2,2,\mathbb{C}$)) we are given that the minimal polynomial of $T$ is $p(z) = (z – 2i)(z + 7)^2$. I want to find the possible Jordan Forms pertaining to this $T$. We know that the characteristic polynomial of $T$ is a polynomial multiple of the minimal polynomial, thus it is either […]

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 & […]

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 […]

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 […]

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 […]

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