Matrix polynomials Polynomial



a matrix polynomial polynomial matrices variables. given ordinary, scalar-valued polynomial







p
(
x
)
=



i
=
0


n




a

i



x

i



=

a

0


+

a

1


x
+

a

2



x

2


+

+

a

n



x

n


,


{\displaystyle p(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}



this polynomial evaluated @ matrix is







p
(
a
)
=



i
=
0


n




a

i



a

i



=

a

0


i
+

a

1


a
+

a

2



a

2


+

+

a

n



a

n


,


{\displaystyle p(a)=\sum _{i=0}^{n}{a_{i}a^{i}}=a_{0}i+a_{1}a+a_{2}a^{2}+\cdots +a_{n}a^{n},}



where identity matrix.


a matrix polynomial equation equality between 2 matrix polynomials, holds specific matrices in question. matrix polynomial identity matrix polynomial equation holds matrices in specified matrix ring mn(r).








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