## 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

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