The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. Specifically, if f(λ) is the characteristic polynomial of a matrix A, then f(A) is the zero matrix. This allows any power of a matrix to be expressed in terms of lower powers. The theorem has applications in calculating functions of matrices and in fields like control systems and electrical circuits.

1. Cayley–Hamilton Theorem
PRESENTED BY
MD. RABIUS SANI
BIPLOB SARKER
FAHMIDA HOSSAIN
BAPPY DAS
NILA

2. About Cayley-Hamilton
Arthur Cayley:(16 August 1821
– 26 January 1895) was a British
Mathematician.

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. 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. 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. Example of Cayley – Hamilton Theorem
Example, let
.
Its characteristic polynomial is given by
The Cayley–Hamilton theorem claims that, if we define
Then

7. Cayley - Hamilton theorem is useful to find
 1.Inverse & power of matrix
 2.only inverse of matrix.
 3.only power of matrix.

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. Thank To All..