>>834 行列式ではなく固有値 トレース（対角成分の和）＝固有値の和 でマルコフ行列だから固有値１を持つのはすぐわかるから .90+.85=1+λが成り立つ これを解いてλ=.75 マルコフ行列（列を足して１になる非負行列）なら固有値1を持つよ 証明は解りやすいのは例えばこれ A Markov matrix A always has an eigenvalue 1. All other eigenvalues are in absolute value smaller or equal to 1. Proof. For the transpose matrix AT , the sum of the row vectors is equal to 1. The matrix AT therefore has the eigenvector [111......1]T. Because A and AT have the same determinant also A . In and AT . In have the same determinant so that the eigenvalues of A and AT are the same. With AT having an eigenvalue 1 also A has an eigenvalue 1. Assume now that v is an eigenvector with an eigenvalue || > 1. Then Anv = ||nv has exponentially growing length for n ! 1. This implies that there is for large n one coefficient [An]ij which is larger than 1. But An is a stochastic matrix (see homework) and has all entries 1. The assumption of an eigenvalue larger than 1 can not be valid.
