Cayley-Hamilton Theorem Definition, Equation & Example

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  • 0:03 Definition of the…
  • 0:27 How the Theorem Works
  • 3:24 History of the Theorem
  • 6:18 Example 1
  • 9:29 Example 2
  • 10:53 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Finding powers of a matrix and determining the inverse of a matrix are applications of the Cayley-Hamilton theorem. In this lesson, the Caley-Hamilton theorem is defined, verified, and applied.

Definition of the Cayley-Hamilton Theorem

Brother and sister, Matt and Poly, have been studying math. Matt enjoys matrices while Poly likes polynomials. The siblings are surprised to learn about polynomials of matrices. Specifically, the Cayley-Hamilton theorem shows how a special polynomial of a matrix is always equal to 0. In this lesson, we define and give examples of this theorem.

How the Theorem Works

Matt and Poly have encountered the square matrix, identity matrix, determinant, and characteristic polynomial. If any of this is unfamiliar, don't worry. The examples will help.

An identity matrix is a square matrix with 1s along the main diagonal and 0s everywhere else. A square matrix has an equal number of rows and columns. This lesson deals exclusively with square matrices (as does the Cayley-Hamilton theorem). Here is a 2-by-2 (2 rows and 2 columns) identity matrix:


2by2_identity_matrix


Here's a 3-by-3 identity matrix:


3by3_identity_matrix


How about multiplying the identity matrix by a number? To keep things general, the number is the variable λ. So λI for a 2-by-2 case is:


lambda_I


λ multiplies each entry of the matrix: λ times 1 is λ; λ times 0 is 0.

For now, use letters to define a 2-by-2 matrix A:


A


There are just two more steps for the characteristic polynomial. First, Matt calculates A - λI:


A-lambdaI


Subtracting one matrix from another involves subtracting the terms at the same locations. The first row, first column of A minus the first row, first column of λI results in a-λ in the first row, first column of the result. Same idea applies for the rest of the subtraction.

Last step is calculating the determinant. For the matrix A - λI, the determinant is (a - λ)(d - λ) - cb. There are more general ways to calculate the determinant for larger matrices but for a 2-by-2 matrix, the product of the terms along the main diagonal minus the product of the terms along the other diagonal is the recipe.


characteristic_polynomial


Poly expands and collects terms. The (a - λ)(d - λ) - cb becomes λ2 - (a + d) λ + ad - cb. This result is called the characteristic polynomial and is labeled p(λ) and is the determinant of the A-λ matrix where the identity matrix I has 1s along the main diagonal and 0s everywhere else.

History of the Theorem


p(A)


In the mid 1800s, British mathematician Arthur Cayley and Irish physicist William Rowan Hamilton discovered and proved an amazing theorem. Recall p(λ) is a polynomial in λ, and the p(λ) equation is based on the terms in the matrix A. The Cayley-Hamilton theorem states if λ is replaced by A, p(A) is equal to zero.

An important detail is the identity matrix I multiplying the ad - cb term so all the terms are matrices.

Time for a numerical example:


numerical_A


Then,


A^2


A times A uses matrix multiplication, which is a row times a column operation. If this is your first time multiplying matrices, the example may or may not be clear enough for you. You are always invited to check out other lessons on matrix operations.

Let's also compute (a + d)A:


(a+d)A


ad - cb = 1(4) - 3(2) = 4 - 6 = -2 so:


(ad-cd)I


Substituting these calculations into p(A):


p(A)=0


The theorem is verified for a 2-by-2 matrix example! This thrills the siblings but they want more. How about some applications?

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