Determine all possible Jordan canonical forms for a linear operator L whose characteristic polynomial is

\( P(\lambda)=\bigg(\lambda-3\bigg)^3\bigg(\lambda-4\bigg)^2 \)

Solutions

Expert Solution

solution

we have the characteristic polynomial is

\( P(\lambda)=\bigg(\lambda-3\bigg)^3\bigg(\lambda-4\bigg)^2 \)

\( \implies am(3)=3,am(4)=2\hspace{2mm}so,\hspace{2mm}gm(3)=1,2,3 \)

and \( gm(4)=1,2 \)

By the order of decreasing of dimension of blocks matrix corresponding to the same eigenvalues.

The possible Jordan matrices are :

\( J_1=\begin{pmatrix}3&0&0&0&0\\ 0&3&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&0\\ 0&0&0&0&4\end{pmatrix}\: \) \( J_2=\:\begin{pmatrix}3&1&0&0&0\\ 0&3&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&0\\ 0&0&0&0&4\end{pmatrix}\: \)

\( J_3=\begin{pmatrix}3&1&0&0&0\\ 0&3&1&0&0\\ 0&0&3&0&0\\ 0&0&0&4&0\\ 0&0&0&0&4\end{pmatrix} \) \( J_4=\:\begin{pmatrix}3&1&0&0&0\\ 0&3&1&0&0\\ 0&0&3&0&0\\ 0&0&0&4&1\\ 0&0&0&0&4\end{pmatrix} \)

\( J_5=\:\begin{pmatrix}3&1&0&0&0\\ 0&3&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&1\\ 0&0&0&0&4\end{pmatrix} \) \( J_6=\:\:\begin{pmatrix}3&0&0&0&0\\ 0&3&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&1\\ 0&0&0&0&4\end{pmatrix} \)

Answer

The possible Jordan matrices are :

\( J_1=\begin{pmatrix}3&0&0&0&0\\ 0&3&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&0\\ 0&0&0&0&4\end{pmatrix}\: \) \( J_2=\:\begin{pmatrix}3&1&0&0&0\\ 0&3&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&0\\ 0&0&0&0&4\end{pmatrix}\: \)

\( J_3=\begin{pmatrix}3&1&0&0&0\\ 0&3&1&0&0\\ 0&0&3&0&0\\ 0&0&0&4&0\\ 0&0&0&0&4\end{pmatrix} \) \( J_4=\:\begin{pmatrix}3&1&0&0&0\\ 0&3&1&0&0\\ 0&0&3&0&0\\ 0&0&0&4&1\\ 0&0&0&0&4\end{pmatrix} \)

\( J_5=\:\begin{pmatrix}3&1&0&0&0\\ 0&3&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&1\\ 0&0&0&0&4\end{pmatrix} \) \( J_6=\:\:\begin{pmatrix}3&0&0&0&0\\ 0&3&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&1\\ 0&0&0&0&4\end{pmatrix} \)

SUN BUNRA answered 2 years ago

Next > < Previous

Related Solutions

Determine all possible Jordan canonical forms J for a matrix of order 6 whose minimal polynomial is

Determine all possible Jordan canonical forms J for a matrix of order 6 whose minimal polynomial is \( m(\lambda)=\bigg(\lambda-1\bigg)^3\bigg(\lambda-3\bigg)^2 \)

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.

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=\( \begin{pmatrix}2&1&0&0\\ 0&2&0&0\\ 0&0&3&0\\ 0&0&0&3\end{pmatrix} \)

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.

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=\( \begin{pmatrix}2&0&0&0\\ 0&2&0&0\\ 0&0&1&0\\ 0&0&0&3\end{pmatrix} \)

Find the characteristics and the minimal polynomial of the following matrices over R, then deduce the their corresponding Jordan Canonical Form J.

Find the characteristics and the minimal polynomial of the following matrices over \( \mathbb{R} \), then deduce the their corresponding Jordan Canonical Form J. A=\( \begin{pmatrix}1&-1&-1\\ 0&0&-1\\ 0&1&2\end{pmatrix}\: \)

Find the characteristics and the minimal polynomial of the following matrices over R , then deduce the their corresponding Jordan Canonical Form J.

Find the characteristics and the minimal polynomial of the following matrices over \( \mathbb{R} \), then deduce the their corresponding Jordan Canonical Form J. B=\( \begin{pmatrix}2&-1&-1&2\\ 0&1&-1&2\\ 2&-5&-1&6\\ 1&-3&-2&6\end{pmatrix} \)

Find the characteristics and the minimal polynomial of the following matrices over R, then deduce their corresponding Jordan Canonical Form J.

Find the characteristics and the minimal polynomial of the following matrices over R, then deduce their corresponding Jordan Canonical Form J. C=\( \begin{pmatrix}1&1&0&0\\ -1&-1&1&0\\ 0&1&1&0\\ -1&-1&1&1\end{pmatrix} \)

Problem 3. An isometry between inner-product spaces V and W is a linear operator L in...

Problem 3. An isometry between inner-product spaces V and W is a linear operator L in B (V ,W) that preserves norms and inner-products. If x, y in V and if L is an isometry, then we have <L(x),L(y)>_W = <x, y>_V . Suppose that V and W are both real, n-dimensional inner-product spaces. Thus the scalar field for both is R and both of them have a basis consisting of n elements. Show that V and W are isometric...

a) For the following polynomial; a. Use the Rational Zero Test to list all possible rational...

a) For the following polynomial; a. Use the Rational Zero Test to list all possible rational roots b. Use Descartes Rule of Signs to provide the possible numbers of positive and negative real roots c. Factor the polynomial completely. ? 3 + 4? 2 + 9? + 36 b) For the following polynomial; d. Use the Rational Zero Test to list all possible rational roots e. Use Descartes Rule of Signs to provide the possible numbers of positive and negative...

Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution,...

Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions (if any exist). i. x1+ x2 +x4 = 3 x2 + x3 = 4 x1 + 2x2 + x3 + x4 = 8 ii. x1 + 2x2 + x3 = 4 x1 + 2x2 = 6 iii. x1 + x2 =1 2x1 + x2=3 3x1 + 2x=...

Determine the eigenvalues and eigenfunctions of the following operator (assume σ(x) ≡ 1): L(u) = u''−2u...

Determine the eigenvalues and eigenfunctions of the following operator (assume σ(x) ≡ 1): L(u) = u''−2u x ∈ (−1,1) with periodic boundary conditions u(−1) = u(1), u'(−1) = u'(1). Box your ﬁnal answer

Subjects