# What is state Cayley-Hamilton theorem?

## What is state Cayley-Hamilton theorem?

In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation. The theorem holds for general quaternionic matrices.

What is the use of Cayley-Hamilton theorem?

Cayley-Hamilton theorem can be used to prove Gelfand’s formula (whose usual proofs rely either on complex analysis or normal forms of matrices). Let A be a d×d complex matrix, let ρ(A) denote spectral radius of A (i.e., the maximum of the absolute values of its eigenvalues), and let ‖A‖ denote the norm of A.

### How do you solve the Cayley-Hamilton theorem?

p(t)=det(A−tI)=[1−t113−t]=t2−4t+2. Then the Cayley-Hamilton theorem says that the matrix p(A)=A2−4A+2I is the 2×2 zero matrix. In fact, we can directly check this: p(A)=A2−4A+2I=[1113][1113]−4[1113]+2[1001]=[24410]+[−4−4−4−12]+[2002]=[0000].

How do you find the 8 using Cayley-Hamilton theorem?

Given that P(t)=t4−2t2+1, the Cayley-Hamilton Theorem yields that P(A)=O, where O is 4 by 4 zero matrix. Then O=A4−2A2+I⟺A4=2A2−I⟹A8=(2A2−I)2. A8=4A4−4A2+I=4(2A2−I)−4A2+I=4A2−3I.

#### How do I get to Cayley Hamilton?

To apply the Cayley-Hamilton theorem, we first determine the characteristic […] Find All the Eigenvalues of Power of Matrix and Inverse Matrix Let A=[3−124−10−2−15−1]. Then find all eigenvalues of A5. If A is invertible, then find all the eigenvalues of A−1.

Is Cayley Hamilton theorem important?

This theorem is used all over in linear algebra. One can easily find inverse of a matrix using Cayley Hamilton theorem. It also plays an important role in solving ordinary differential equations[2]. This theorem is quite useful in physics also.

## How do you find the 100 Cayley Hamilton theorem?

Use the Cayley-Hamilton Theorem to Compute the Power A100

1. (b) Let.
2. Note that the product of all eigenvalues of A is the determinant of A.
3. To use the Cayley-Hamilton theorem, we first need to determine the characteristic polynomial p(t)=det(A−tI) of A.
4. Then the Cayley-Hamilton theorem yields that.

Does every square matrix satisfies its characteristic equation?

In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.