# 4.3 - Section 4.3 EIGENVALUES OF SPECIAL MATRICES Example...

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Section 4.3 EIGENVALUES OF SPECIAL MATRICES Example: Find the eigenvalues. 14 21 A  =   AI λ −= det( ) 0 = () 11 8 0 λλ −− = 2 12 8 0 −+ = 2 27 0 =

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2 4( )( ) 2 bb a c a λ −± = 24 4 ( 1 ) ( 7 ) 2 ±− = 23 2 4 2 11 2 2 22 ± == ± = ± Theorem: MATRICES WITH REAL NUMBER ENTRIES MAY HAVE COMPLEX EIGENVALUES. Example: Let 12 A  =   Find the eigenvalues. Solution: Set det A − λ I () = 0 and solve for λ .
12 22 AI λ  −=  −−  (1 )(2 ) 4 0 λλ =− + = 2 23 4 + + 2 36 =−+ 3 9-4(1)(6) = 2 ± 3- 1 5 2 ± = 31 5 2 i ± = For this class, make sure you know that 1 i . This is an "imaginary" number. Complex numbers are in the form a+bi . a is the real part and bi is the imaginary part.

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Several questions will ask for ___+___i and you’ll have to fill in the blanks. Theorem: FOR A REAL MATRIX, COMPLEX EIGENVALUES OCCUR IN CONJUGATE PAIRS; THAT IS, IF a + bi IS AN EIGENVALUE THEN SO IS a - bi .
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4.3 - Section 4.3 EIGENVALUES OF SPECIAL MATRICES Example...

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