Linear Algebra

242 Chapter Three. Maps Between Spacesapply ˆT,((5 − √ 3)/6 (3 + 2 √ ) ( )3)/3(1 + √ 1√3)/6 3/3 1ˆB, ˆDand check it against ˆD11 + 3 √ (3 −1·6 0)+ 1 + 3√ 36and it gives the same outcome as above.ˆB((11 + 3 √ )3)/6=(1 + 3 √ 3)/6( ) (2 (−3 + √ )3)/2· =3 (1 + 3 √ 3)/22.2 Example We may make the matrix simpler by changing bases. On R 3 themap⎛ ⎞ ⎛ ⎞x y + z⎜ ⎟⎝y⎠↦−→t ⎜ ⎟⎝x + z⎠z x + yis represented with respect to the standard basis in this way.⎛ ⎞0 1 1⎜ ⎟Rep E3 ,E 3(t) = ⎝1 0 1⎠1 1 0Represented with respect to⎛ ⎞ ⎛ ⎞ ⎛1 1⎜ ⎟ ⎜ ⎟ ⎜1⎞⎟B = 〈 ⎝−1⎠ , ⎝ 1⎠ , ⎝1⎠〉0 −2 1gives a matrix that is diagonal.⎛⎞−1 0 0⎜⎟Rep B,B (t) = ⎝ 0 −1 0⎠0 0 2Naturally we usually prefer basis changes that make the representation easierto understand. We say that a map or matrix has been diagonalized when itsrepresentation is diagonal with respect to B, B, that is, with respect to equalstarting and ending bases. In Chapter Five we shall see which maps and matricesare diagonalizable. In the rest of this subsection we consider the easier casewhere representations are with respect to B, D, which are possibly differentstarting and ending bases. Recall that the prior subsection shows that a matrixchanges bases if and only if it is nonsingular. That gives us another version ofthe above arrow diagram and equation (∗) from the start of this subsection.2.3 Definition Same-sized matrices H and Ĥ are matrix equivalent if there arenonsingular matrices P and Q such that Ĥ = PHQ.ˆD