• If λ = eigenvalue, then x = eigenvector (an eigenvector is always associated with an eigenvalue) Eg: If L(x) = 5x, 5 is the eigenvalue and x is the eigenvector. :5/ . Question: If λ Is An Eigenvalue Of A Then λ − 7 Is An Eigenvalue Of The Matrix A − 7I; (I Is The Identity Matrix.) 2. In case, if the eigenvalue is negative, the direction of the transformation is negative. This eigenvalue is called an inﬁnite eigenvalue. Observation: det (A – λI) = 0 expands into a kth degree polynomial equation in the unknown λ called the characteristic equation. If x is an eigenvector of the linear transformation A with eigenvalue λ, then any vector y = αx is also an eigenvector of A with the same eigenvalue. In fact, together with the zero vector 0, the set of all eigenvectors corresponding to a given eigenvalue λ will form a subspace. B = λ I-A: i.e. Therefore, λ 2 is an eigenvalue of A 2, and x is the corresponding eigenvector. Here is the most important definition in this text. An application A = 10.5 0.51 Given , what happens to as ? If λ = 1, the vector remains unchanged (unaffected by the transformation). Eigenvalues so obtained are usually denoted by λ 1 \lambda_{1} λ 1 , λ 2 \lambda_{2} λ 2 , …. The eigenvalue λ is simply the amount of "stretch" or "shrink" to which a vector is subjected when transformed by A. The ﬁrst column of A is the combination x1 C . then λ is called an eigenvalue of A and x is called an eigenvector corresponding to the eigen-value λ. detQ(A,λ)has degree less than or equal to mnand degQ(A,λ)

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