MadeEasy Test Series: Linear Algebra - Eigen Value

Question

A 3 X 3 matrix P has 3 eigen values $-1 , 0.5 , 3$
What will be eigen values of $P^{2} + 2P + I$

Comments

According to Cayley Hamilton theorem: Every Square matrix satisfies its own characteristic equation.

                  So$, P=\lambda I$

Given that $\lambda_{1}=0,\lambda_{2}=0.5,\lambda_{3}=3$

$P^{2}+2P+I=(\lambda I)^{2}+2(\lambda I)+I=\lambda ^{2}+2\lambda +1$

Put $\lambda=0$

$\lambda^{2}+2\lambda +1 = 0+0+1=1$

Put $\lambda=0.5$

$\lambda^{2}+2\lambda_{3} +1=(0.5)^{2}+2(0.5)+1=0.25+1+1=2.25$

Put $\lambda=3$

$\lambda^{2}+2\lambda +1=(3)^{2}+2(3)+1=9+6+1=16$

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Look here for a good proof (Page 7), https://www.adelaide.edu.au/mathslearning/play/seminars/evalue-magic-tricks-handout.pdf

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This is nice pdf thanks!!

Answer 1

k1 = -1, k2 = 0.5, k3 = 3

Let P be a diagonal Matrix..!!

So, the elements of the principal diagonal of P will only be the eigen values of P..!
the square of diagonal matrix will yield in the square of the principal diagonal elements with other entries to be zero..

So, P^(2) + 2P + I>.
Will be having the eigen values as..

K1 = (1 - 2 + 1)       =  0
K2 = (0.25 + 1 + 1)  =  2.25
K3 = (9 + 6 + 1)       =  16