# Find Jordan Canonical Form with Maple

• Maple
• joypav
In summary, the matrix 10x10 is nilpotent, its signature is (0, 5, 0, 3, 0, 2), and its Jordan canonical form is (0, 5, 0, 3, 0, 2).
joypav
Hi all!
I have to show that the matrix 10x10 matrix below is nilpotent, determine its signature, and find its Jordan canonical form.

[-2 , 19/2 , -17/2 , 0 , -13 , 9 , -4 , 7 , -2 , -13]
[15 , -51 , 48 , -8 , 80 , -48 , 19 , -39 , 10 , 74]
[-7 , 34 , -33 , 0 , -50 , 31 , -11 , 27 , -6 , -47]
[1 , -4 , 4 , -1 , 7 , -4 , 1 , -3 , 1 , 6]
[0 , -2 , 2 , 0 , 3 , -2 , 1 , -1 , 0 , 3]
[-11 , 91/2 , -87/2 , 4 , -70 , 42 , -16 , 35 , -8 , -65]
[-1 , 9/2 , -9/2 , 0 , -7 , 4 , -1 , 4 , -1 , -6]
[-9 , 39 , -37 , 2 , -58 , 36 , -14 , 30 , -7 , -55]
[-7 , 32 , -31 , 1 , -48 , 29 , -11 , 25 , -5 , -45]
[5 , -25/2 , 23/2 , -3 , 20 , -12 , 5 , -10 , 3 , 18]

I've shown that the matrix to the sixth power is the zero matrix (so it's nilpotent). And I believe the signature is (0, 5, 0, 3, 0, 2).. if I've done it correctly.

Anyways, I have not yet found the Jordan Canonical form. We are allowed to use Maple but I do not yet have it for my laptop (an old professor of mine said he would get it for me so I don't have to pay). I know there is a command in Maple that will find it for you.
Could someone with Maple (or Matlab) find it for me? I would really appreciate it!

joypav said:
Hi all!
I have to show that the matrix 10x10 matrix below is nilpotent, determine its signature, and find its Jordan canonical form.

[-2 , 19/2 , -17/2 , 0 , -13 , 9 , -4 , 7 , -2 , -13]
[15 , -51 , 48 , -8 , 80 , -48 , 19 , -39 , 10 , 74]
[-7 , 34 , -33 , 0 , -50 , 31 , -11 , 27 , -6 , -47]
[1 , -4 , 4 , -1 , 7 , -4 , 1 , -3 , 1 , 6]
[0 , -2 , 2 , 0 , 3 , -2 , 1 , -1 , 0 , 3]
[-11 , 91/2 , -87/2 , 4 , -70 , 42 , -16 , 35 , -8 , -65]
[-1 , 9/2 , -9/2 , 0 , -7 , 4 , -1 , 4 , -1 , -6]
[-9 , 39 , -37 , 2 , -58 , 36 , -14 , 30 , -7 , -55]
[-7 , 32 , -31 , 1 , -48 , 29 , -11 , 25 , -5 , -45]
[5 , -25/2 , 23/2 , -3 , 20 , -12 , 5 , -10 , 3 , 18]

I've shown that the matrix to the sixth power is the zero matrix (so it's nilpotent). And I believe the signature is (0, 5, 0, 3, 0, 2).. if I've done it correctly.

Anyways, I have not yet found the Jordan Canonical form. We are allowed to use Maple but I do not yet have it for my laptop (an old professor of mine said he would get it for me so I don't have to pay). I know there is a command in Maple that will find it for you.
Could someone with Maple (or Matlab) find it for me? I would really appreciate it!

With GNU Octave (open source version of MatLab):
Code:
A=[[-2 , 19/2 , -17/2 , 0 , -13 , 9 , -4 , 7 , -2 , -13]
[15 , -51 , 48 , -8 , 80 , -48 , 19 , -39 , 10 , 74]
[-7 , 34 , -33 , 0 , -50 , 31 , -11 , 27 , -6 , -47]
[1 , -4 , 4 , -1 , 7 , -4 , 1 , -3 , 1 , 6]
[0 , -2 , 2 , 0 , 3 , -2 , 1 , -1 , 0 , 3]
[-11 , 91/2 , -87/2 , 4 , -70 , 42 , -16 , 35 , -8 , -65]
[-1 , 9/2 , -9/2 , 0 , -7 , 4 , -1 , 4 , -1 , -6]
[-9 , 39 , -37 , 2 , -58 , 36 , -14 , 30 , -7 , -55]
[-7 , 32 , -31 , 1 , -48 , 29 , -11 , 25 , -5 , -45]
[5 , -25/2 , 23/2 , -3 , 20 , -12 , 5 , -10 , 3 , 18]];

jordan(sym(A))
⎡0  1  0  0  0  0  0  0  0  0⎤
⎢                            ⎥
⎢0  0  1  0  0  0  0  0  0  0⎥
⎢                            ⎥
⎢0  0  0  1  0  0  0  0  0  0⎥
⎢                            ⎥
⎢0  0  0  0  1  0  0  0  0  0⎥
⎢                            ⎥
⎢0  0  0  0  0  0  0  0  0  0⎥
⎢                            ⎥
⎢0  0  0  0  0  0  1  0  0  0⎥
⎢                            ⎥
⎢0  0  0  0  0  0  0  1  0  0⎥
⎢                            ⎥
⎢0  0  0  0  0  0  0  0  0  0⎥
⎢                            ⎥
⎢0  0  0  0  0  0  0  0  0  1⎥
⎢                            ⎥
⎣0  0  0  0  0  0  0  0  0  0⎦

## 1. What is the Jordan Canonical Form?

The Jordan Canonical Form is a way to represent a square matrix in a specific form that makes it easier to analyze and solve problems. It consists of a diagonal matrix with Jordan blocks along the diagonal, where each Jordan block represents a single eigenvalue of the original matrix.

## 2. Why is it important to find the Jordan Canonical Form?

Finding the Jordan Canonical Form allows us to simplify and analyze complex matrices, making it easier to solve problems related to linear algebra and differential equations. It also helps in understanding the properties of a given matrix, such as its eigenvalues and eigenvectors.

## 3. How does Maple help in finding the Jordan Canonical Form?

Maple is a powerful mathematical software that has built-in functions and algorithms to find the Jordan Canonical Form of a given matrix. It allows for efficient and accurate computations, making it a useful tool for scientists and mathematicians.

## 4. What are the steps to find the Jordan Canonical Form using Maple?

The steps to find the Jordan Canonical Form using Maple are as follows:1. Define the matrix using the Matrix() function.2. Use the Eigenvectors() function to find the eigenvectors of the matrix.3. Use the JordanForm() function to compute the Jordan Canonical Form.

## 5. Can Maple find the Jordan Canonical Form for any size of matrix?

Yes, Maple can find the Jordan Canonical Form for any size of matrix. However, as the size of the matrix increases, the computation time may also increase. It is recommended to use Maple's parallel processing capabilities for larger matrices to speed up the computation process.

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