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How do you find the Cayley Hamilton theorem of a matrix?

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How do you find the Cayley Hamilton theorem of a matrix?

To use the Cayley-Hamilton theorem, we first compute the characteristic polynomial p(t) of […] 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.

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.

What is Cayley Hamilton theorem example?

Cayley Hamilton Theorem Example Suppose a matrix is given as A = [1234] [ 1 2 3 4 ] . The characteristic polynomial is λ2−5λ−2 λ 2 − 5 λ − 2 . Use the matrix A in place of the variable to get p(A) = A2 – 5A – 2I = [1234]2−5[1234]−2[1001] [ 1 2 3 4 ] 2 − 5 [ 1 2 3 4 ] − 2 [ 1 0 0 1 ] .

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

What is Cayley-Hamilton theorem example?

What is Cayley-Hamilton rule?

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.

What is Cayley Hamilton method?

How do you find the 100 matrix by 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.

What is an example of Cayley Hamilton theorem?

Example of Cayley-Hamilton Theorem 1.) 1 x 1 Matrices For 1 x 1 matrix A (a 1,1) the characteristic polynomial is given by and so p (A) = (a) – (a 1,1) = 0 is obvious.

Does Cayley’s theorem hold for all quaternionic matrices?

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The theorem holds for broad quaternionic matrices. Cayley in 1858 said it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case. The general case was first verified by Frobenius in 1878.

What is the origin of the Hamiltonian theorem?

The theorem was first proved in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton. This parallels to the special case of certain real 4 × 4 real or 2 × 2 complex matrices. The theorem holds for broad quaternionic matrices. Cayley in 1858 said it for 3 × 3 and smaller matrices,…

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