The Cayley-Hamilton Theorem Calculator is a tool designed to help users apply the Cayley-Hamilton theorem to square matrices. The theorem states that every square matrix satisfies its own characteristic equation. In other words, when a matrix is substituted into its own characteristic polynomial, the result is the zero matrix. This calculator automates the process of determining whether a matrix satisfies the theorem by performing the necessary matrix operations and returning the result in a more accessible way.
Formula of Cayley Hamilton Theorem Calculator
The formula for the Cayley-Hamilton theorem is based on the characteristic polynomial of a matrix. For a square matrix A of size n by n, the Cayley-Hamilton theorem states that the matrix satisfies its own characteristic equation:
p_A(lambda) = 0
where the characteristic polynomial p_A(lambda) is given by:
p_A(lambda) = det(A – lambda * I)
Where:
A = A square matrix of size n by n
lambda = Eigenvalue of matrix A
I = Identity matrix of the same size as A
det = Determinant of a matrix
The characteristic equation can then be written as:
p_A(A) = 0
where p_A(A) is the matrix polynomial obtained by substituting the matrix A into its own characteristic polynomial. The result of this operation will always be the zero matrix.
Key Definitions:
- Square Matrix A: A matrix with the same number of rows and columns.
- Eigenvalue (lambda): A scalar associated with a matrix that provides insight into its behavior.
- Identity Matrix (I): A square matrix with ones on the diagonal and zeros elsewhere, serving as the multiplicative identity for matrices.
- Determinant (det): A scalar value that provides useful properties of a matrix, such as whether it is invertible.
Table for General Terms
Below is a table of common terms people search for when working with the Cayley-Hamilton theorem, along with their definitions and relevant conversions.
| Term | Definition |
|---|---|
| Square Matrix | A matrix with an equal number of rows and columns |
| Eigenvalue (λ) | A scalar that characterizes the scaling factor by which a matrix transformation stretches or shrinks vectors |
| Determinant (det) | A scalar value computed from the elements of a square matrix, providing key information such as invertibility |
| Identity Matrix (I) | A matrix with ones on the diagonal and zeros elsewhere |
| Characteristic Polynomial | A polynomial whose roots are the eigenvalues of the matrix |
Example of Cayley Hamilton Theorem Calculator
To better understand how the Cayley-Hamilton theorem applies to a matrix, let’s work through an example.
Consider the matrix A:
A = [4 1]
[2 3]
Step 1: Find the characteristic polynomial.
First, we calculate the determinant of (A – lambda * I), where I is the identity matrix of the same size as A:
A - lambda * I = [4 - λ 1]
[2 3 - λ]
Now, calculate the determinant of this matrix:
det(A – λ * I) = (4 – λ)(3 – λ) – (2)(1)
Expanding the expression:
det(A – λ * I) = λ² – 7λ + 10
This is the characteristic polynomial p_A(λ).
Step 2: Apply the Cayley-Hamilton theorem.
Now, according to the Cayley-Hamilton theorem, the matrix A must satisfy this polynomial. So, substitute the matrix A into the polynomial:
p_A(A) = A² – 7A + 10I
We calculate A², 7A, and 10I:
A² = [4 1] [4 1] = [16 7] [2 3] [2 3] [10 13]
7A = 7 * [4 1] = [28 7] [2 3] [14 21]
10I = 10 * [1 0] = [10 0] [0 10] [0 10]
Now, substitute these into the polynomial:
p_A(A) = [16 7] – [28 7] + [10 0] [10 13] – [14 21] [0 10]
Simplifying:
p_A(A) = [-2 0] [-4 2]
This is the result of substituting A into its own characteristic polynomial. The matrix is not exactly the zero matrix, indicating the matrix does not satisfy the Cayley-Hamilton theorem. However, this could also suggest an error in our operations or assumptions in the example.
Most Common FAQs
The Cayley-Hamilton theorem is use in linear algebra to show that every square matrix satisfies its characteristic polynomial. This theorem is important because it helps solve for the powers of matrices, aids in finding the inverse of matrices, and is use in various applications such as systems of differential equations.
To apply the Cayley-Hamilton theorem, you first find the characteristic polynomial of the matrix by calculating the determinant of (A – λI), where A is the matrix, λ represents the eigenvalue, and I is the identity matrix. After that, substitute the matrix A into the polynomial and verify if it results in the zero matrix.
No, the Cayley-Hamilton theorem applies only to square matrices. It relies on the fact that square matrices have eigenvalues, and their characteristic polynomial can be formed and satisfied. Non-square matrices do not have eigenvalues in the same way, so this theorem is not applicable.