Matrix Problems

Problem. Given n-unknown functions x1(t), x2(t), . . ., xn(t), suppose 

x′1 = a11x1 + a12x2 + . . . + a1nxn x′2 = a21x1 + a22x2 + . . . + a2nxn .. .. ............... x′n = an1x1 + an2x2 + . . . + annxn,

where aij are constants. Making a change of variable

 x1 = b11y1 + b12y2 + . . . + b1nyn x2 = b21y1 + b22y2 + . . . + b2nyn .. .. ............... xn = bn1y1 + bn2y2 + . . . + bnnyn,

where (bij)n×n is some invertible matrix, determine the equations that the new functions y1(t), y2(t), . . ., yn(t) need to satisfy.

Be sure to write your answer in term of matrices and vectors.

Note: Answers to this problem were explained in detail in lecture today. If you do this problem, you will understand why people spend big effort in linear algebra trying to find an invertible matrix B = (bij) so that

B−1AB

is simpler, where A = (aij) is any given matrix.

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