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# Cayley – hamiltion theorem By Sunny

Cayley-Hamilton Theorem

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### Cayley – hamiltion theorem By Sunny

1. 1. Cayley–Hamilton Theorem PRESENTED BY MD. RABIUS SANI BIPLOB SARKER FAHMIDA HOSSAIN BAPPY DAS NILA
2. 2. About Cayley-Hamilton Arthur Cayley:(16 August 1821 – 26 January 1895) was a British Mathematician.
3. 3. About Sir William Rowan Hamilton Sir William Rowan Hamilton: (3– 4 August 1805 – 2 September 1865) was an Irish physicist, astronomy, and mathematician
4. 4. Cayley-Hamilton Theorem The Cayley - Hamilton Theorem states that a square n*n matrix A satisfies own characteristic equation. Thus, we can express An in terms of a finite set of lower powers of A. This fact leads to a simple way of calculating the value of a function evaluated at the matrix A is: f(λ)=A is λ n + a1 λ n-1 + a2 λ n-2+…………+an-1 λ +an=0 Where I is the nth order unit matrix and 0 is the nth order zero matrix.
5. 5. Proof  The characteristic equation for the matrix A is |λI-A|=0 which can be written as An + a1An-1 + a2An-2+…………+an-1A+anI=0
6. 6. Example of Cayley – Hamilton Theorem Example, let . Its characteristic polynomial is given by The Cayley–Hamilton theorem claims that, if we define Then
7. 7. Cayley - Hamilton theorem is useful to find  1.Inverse & power of matrix  2.only inverse of matrix.  3.only power of matrix.
8. 8. Uses of Cayley-Hamilton Theorem in EEE  The Cayley -Hamilton theorem and its generalizations have been used in control systems, electrical circuits, systems with delays, singular systems. specially in DC circuit
9. 9. Thank To All..