Let \( A\in M_n(\mathbb{R})\hspace{2mm} \) and\( \hspace{2mm} m_A(\lambda)\hspace{2mm} \) be its minimal polynomial. Let f be a polynomial satisfies\( \hspace{2mm}f(A) = 0. \hspace{2mm} \)Show that\( \hspace{2mm} f(\lambda) \hspace{2mm} \)is divisible by\( \hspace{2mm} m_A(\lambda). \)
In: Advanced Math
Determine all possible Jordan canonical forms J for a matrix of order 6 whose minimal polynomial is
\( m(\lambda)=\bigg(\lambda-1\bigg)^3\bigg(\lambda-3\bigg)^2 \)
In: Advanced Math
Determine all possible Jordan canonical forms for a linear operator L whose characteristic polynomial is
\( P(\lambda)=\bigg(\lambda-3\bigg)^3\bigg(\lambda-4\bigg)^2 \)
In: Advanced Math
The following are the Jordan Canonical Form of linear transformations. Find the characteristic polynomial, minimal polynomials, the algebraic multiplicity, geometric multiplicity and the index of each of the eigenvalues of L.
B=\( \begin{pmatrix}2&1&0&0\\ 0&2&0&0\\ 0&0&3&0\\ 0&0&0&3\end{pmatrix} \)
In: Advanced Math
The following are the Jordan Canonical Form of linear transformations. Find the characteristic polynomial, minimal polynomials, the algebraic multiplicity, geometric multiplicity and the index of each of the eigenvalues of L.
A=\( \begin{pmatrix}2&0&0&0\\ 0&2&0&0\\ 0&0&1&0\\ 0&0&0&3\end{pmatrix} \)
In: Advanced Math
Find the characteristics and the minimal polynomial of the following matrices over R, then deduce their corresponding Jordan Canonical Form J.
C=\( \begin{pmatrix}1&1&0&0\\ -1&-1&1&0\\ 0&1&1&0\\ -1&-1&1&1\end{pmatrix} \)
In: Advanced Math
Find the characteristics and the minimal polynomial of the following matrices over \( \mathbb{R} \), then deduce the their corresponding Jordan Canonical Form J.
B=\( \begin{pmatrix}2&-1&-1&2\\ 0&1&-1&2\\ 2&-5&-1&6\\ 1&-3&-2&6\end{pmatrix} \)
In: Advanced Math
Find the characteristics and the minimal polynomial of the following matrices over \( \mathbb{R} \), then deduce the their corresponding Jordan Canonical Form J.
A=\( \begin{pmatrix}1&-1&-1\\ 0&0&-1\\ 0&1&2\end{pmatrix}\: \)
In: Advanced Math
Find all eigenvalues and their corresponding eigenspaces of the following matrix. B=\( \begin{pmatrix}2&-3&1\\ 1&-2&1\\ 1&-3&2\end{pmatrix} \)
In: Advanced Math
Find all eigenvalues and their corresponding eigenspaces of the following matrix. A.\( \begin{pmatrix}3&0\\ 1&2\end{pmatrix} \)
In: Advanced Math
show that for some ring R, the equality a^2−b^2=(a−b)(a+b) holds ∀a,b∈R if and only if R is commutative.
In: Advanced Math
Show that the relation 'a R b if and only if a−b is an even integer defined on the Z of integers is an equivalence relation.
In: Advanced Math
Prove that in a finite cyclic group, each subgroup has size dividing the size of the group. Conversely, given a positive divisor of the size of the group, there is a subgroup of that size.
In: Advanced Math
Let f be measurable and B a Borel set. Then f-1[B] is a measurable set. [Hint: The class of sets for which f-1[E] is measurable is a σ-algebra.
In: Advanced Math
Let G be finite with |G| > 1. If Aut(G) acts transitively on G − {e} then G ∼= (Z/(p))n for some prime p.
In: Advanced Math