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Mathematical Methods by Pearson
book

Mathematical Methods by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
437 pages
17h 41m
English
Pearson India
Content preview from Mathematical Methods by Pearson
Eigenvalues and Eigenvectors 2-9
B
0
AB
1
= p
1
I
AB
0
= p
0
I
Multiplying these equations by A
n
, A
n1
, …, A; I,
respectively, and adding we get
0 = A
n
+ p
n1
A
n1
+ p
n2
A
n2
+ p
1
A + p
0
I (2.11)
This proves the theorem.
2.5.1 Inverse of a Matrix by
Cayley–Hamilton Theorem
By the Cayley–Hamilton Theorem every square
matrix A satisfies its characteristic equation
A
n
+ p
n1
A
n1
+ p
n2
A
n2
+
+ p
1
A + p
0
I = 0
(2.12)
A
1
exists A is nonsingular
det A = |A| = D(0) = p
0
0
Multiplying (2.12) by A
1
112
12 10
12 1
110
112
11
0
()0
0
1
nn n
nn
nn
n
nn
n
AA pA pA pApI
ApA pIpA
AApApI
p
−−−
−−
−−
−−−
++ +++=
⇒+ +++=
⎡⎤
⇒=− + ++
⎣⎦
(2.13)
This result gives us the in ...
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Publisher Resources

ISBN: 9781299446557Publisher Website