characteristic polynomial

Definitions

1. The polynomial produced from a given square matrix by first subtracting the appropriate identity matrix multiplied by an indeterminant and then calculating the determinant.
2. A polynomial P(r) corresponding to a homogeneous, linear, ordinary differential equation P(D) y = 0 where D is a differential operator (with respect to a variable t, if y is a function of t).

Equivalents

Examples

“The characteristic polynomial of #92;textstyle#92;left(#92;begin⟳#123;array#125;#123;cc#125;1amp;4#92;#92;3amp;-5#92;end⟳#123;array#125;#92;right) is #92;textstyle#92;left#92;vert#92;begin⟳#123;array#125;#123;cc#125;1-xamp;4#92;#92;3amp;-5-x#92;end⟳#123;array#125;#92;right#92;vert#61;x²#43;4x-17.”

“The characteristic polynomial of a 2#92;times 2 matrix M is #92;lambda²-#92;mbox#123;tr#125;(M)#92;lambda#43;#92;mbox#123;det#125;(M), where #92;mbox#123;tr#125;(M) denotes the trace of M and #92;mbox#123;det#125;(M) denotes the determinant of M.”

“The characteristic polynomial of a 3#92;times 3 matrix M is -#92;lambda³#43;#92;mbox#123;tr#125;(M)#92;lambda²-#92;mbox#123;tr#125;(#92;mbox#123;adj#125;(M))#92;lambda#43;#92;mbox#123;det#125;(M), where #92;mbox#123;adj#125;(M) denotes the adjugate of M.”

CEFR level