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Chapter 9: Problem 8

Prove the Cayley-Hamilton Theorem 9.2: Every matrix \(A\) is a root of its characterstic polynomial \(\Delta(t)\)

Short Answer

Expert verified
To prove the Cayley-Hamilton theorem, follow these steps: 1. Define the characteristic polynomial 螖(t) using the determinant of (tI - A). 2. Compute the determinant of (tI - A) to obtain the polynomial 螖(t). 3. Replace the scalar variable 't' in the characteristic polynomial with matrix A. 4. Perform matrix operations to simplify the expression and verify that the result is the zero matrix, thus proving the theorem.

Step by step solution

01

Define the characteristic polynomial

First, let's define the characteristic polynomial of matrix A. The characteristic polynomial, 螖(t), is given by the determinant of (tI - A), where t is a scalar variable, I is the identity matrix of the same size as A, and A is the given matrix.
02

Compute the determinant of (tI - A)

Now, we need to compute the determinant of (tI - A). This is a polynomial in t, which we will call 螖(t). Remember, the goal is to show that A is a root of 螖(t), meaning that when we replace t with A, the resulting matrix equation equals the zero matrix.
03

Replace 't' with matrix A in the characteristic polynomial

In this step, we need to replace the scalar variable 't' in the characteristic polynomial with matrix A. Keep in mind that operations with matrices follow different rules than those with scalars. You may need to perform operations like matrix multiplication or computing the power of a matrix, depending on the characteristic polynomial obtained in Step 2.
04

Verify that the result is the zero matrix

After replacing t with matrix A in the characteristic polynomial, perform all the necessary matrix operations to simplify the expression. The final result must be the zero matrix of the same size as A. If this is the case, then we have successfully proven the Cayley-Hamilton theorem for matrix A.

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Most popular questions from this chapter

For each of the following matrices, find all eigenvalues and a maximum set \(S\) of linearly independent eigenvectors: (a) \(A=\left[\begin{array}{ccc}1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4\end{array}\right]\) (c) \(C=\left[\begin{array}{rrr}1 & 2 & 2 \\ 1 & 2 & -1 \\ -1 & 1 & 4\end{array}\right]\) (b) \(B=\left[\begin{array}{ccc}3 & -1 & 1 \\ 7 & -5 & 1 \\ 6 & -6 & 2\end{array}\right]\) Which matrices can be diagonalized, and why?

Let \(m(t)\) be the minimal polynomial of an \(n\) -square matrix \(A\). Prove that the characteristic polynomial \(\Delta(t)\) of \(A\) divides \([m(t)]^{n}\)

Show that matrices \(A\) and \(A^{T}\) have the same eigenvalues. Give an example of a \(2 \times 2\) matrix \(A\) where \(A\) and \(A^{T}\) have different eigenvectors.

Suppose \(f(t)\) is an irreducible monic polynomial for which \(f(A)=0\) for a matrix \(A .\) Show that \(f(t)\) is the minimal polynomial of \(A\)

Using variables \(s\) and \(t,\) find an orthogonal substitution that diagonalizes cach of the following quadratic forms: (a) \(q(x, y)=4 x^{2}+8 x y-11 y^{2}\) (b) \(q(x, y)=2 x^{2}-6 x y+10 y^{2}\)
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