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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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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.

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.

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:

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

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

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

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

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

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 Î»I results in a-Î» 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 - Î»I, the determinant is (a - Î»)(d - Î») - 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.

Poly expands and collects terms. The (a - Î»)(d - Î») - cb becomes Î»2 - (a + d) Î» + ad - cb. This result is called the **characteristic polynomial** and is labeled p(Î») and is the determinant of the A-Î» matrix where the identity matrix I has 1s along the main diagonal and 0s everywhere else.

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

Then,

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:

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

Substituting these calculations into p(A):

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

For this first example, let's use Using the Cayley-Hamilton theorem to find the inverse of A. The **matrix inverse** is written A-1. Some facts about A-1:

- A-1 A = AA-1 = I; a matrix times its inverse equals the identity matrix I
- A-1 An = An-1; for example, A-1 A2 = A1, which equals A
- A-1 â‰ 1/A; the matrix inverse does not equal the reciprocal of the matrix

To find A-1, start by computing det(A - Î»I):

This is the characteristic polynomial p(Î») = Î»2 - 5Î» - 2. Using the Caley-Hamilton theorem, p(A) = 0. Thus, A2 - 5A - 2I = 0.

Multiply through by A-1:

- A-1 A2 is A
- A-1 times 5A is 5 A-1 A, which is 5I
- A-1 times 2I is 2 A-1
- A-1 times 0 is 0

A2 - 5A - 2I = 0. We have A - 5I - 2A-1 = 0. Solving for A-1:

As a check, multiplying A by its inverse must give the identity matrix I:

The math siblings have checked the inverse of A. Time for one more application.

For this next example, let's try using the Cayley-Hamilton theorem to find An for n=3.

From A2 - 5A - 2I = 0, multiply through by A:

- A times A2 is A3
- A times 5A is 5A2
- A times 2I is 2A
- A times 0 is 0

Thus, A2 - 5A - 2I = 0 becomes A3 - 5A2 - 2A = 0.

Solving for A3 gives A3 = 5 A2 + 2A. Using the already computed A2 and substituting:

We can iteratively calculate higher and higher powers of A. To find An, all we need are An-1 and An-2.

The siblings are indeed very happy!

The **Cayley-Hamilton theorem** deals with square matrices and shows how a special polynomial of a matrix is always equal to 0. A **square matrix** has an equal number of rows and columns. From the matrix A and the variable Î», the characteristic polynomial is computed. The **characteristic polynomial**, labeled p(Î») is the determinant of the A - Î»I matrix where the **identity matrix** I has 1s along the main diagonal and 0s everywhere else. Substituting A for Î» in p(Î») gives p(A). The Caley-Hamilton theorem states p(A) = 0. The resulting equation can be used to find the **matrix inverse** and iteratively to find powers of a matrix.

It's possible that, upon finishing this lesson, you could confidently do the following:

- Make the distinction between square matrices, characteristic polynomials and identify matrices
- Explain how the Cayley-Hamilton theory works
- Apply the Cayley-Hamilton theory

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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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