Linear Algebra

346 Chapter Five. SimilarityI.2 Complex RepresentationsRecall the definitions of the complex number additionand multiplication.(a + bi) + (c + di) = (a + c) + (b + d)i(a + bi)(c + di) = ac + adi + bci + bd(−1)= (ac − bd) + (ad + bc)i2.1 Example For instance, (1 − 2i) + (5 + 4i) = 6 + 2i and (2 − 3i)(4 − 0.5i) =6.5 − 13i.Handling scalar operations with those rules, all of the operations that we’vecovered for real vector spaces carry over unchanged.2.2 Example Matrix multiplication is the same, although the scalar arithmeticinvolves more bookkeeping.( ) ( )1 + 1i 2 − 0i 1 + 0i 1 − 0ii −2 + 3i 3i −i( )(1 + 1i) · (1 + 0i) + (2 − 0i) · (3i) (1 + 1i) · (1 − 0i) + (2 − 0i) · (−i)=(i) · (1 + 0i) + (−2 + 3i) · (3i) (i) · (1 − 0i) + (−2 + 3i) · (−i)( )1 + 7i 1 − 1i=−9 − 5i 3 + 3iEverything else from prior chapters that we can, we shall also carry overunchanged. For instance, we shall call this⎛ ⎞ ⎛ ⎞1 + 0i 0 + 0i0 + 0i〈 ⎜⎝⎟. ⎠ , . . . , 0 + 0i⎜⎝⎟. ⎠ 〉0 + 0i 1 + 0ithe standard basis for C n as a vector space over C and again denote it E n .