Summary of common concepts and problems of linear algebra in postgraduate entrance examination

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1. A matrix has many eigenvalues and eigenvectors , Any eigenvalue λ And the corresponding eigenvector , All satisfied with Aα=λα！ Instead of foolishly putting all λ and α combined , Because that's an orthogonal transformation . Don't confuse ！
2. A zero vector cannot be said to be an eigenvector , Look at the definition ！α Is a nonzero column vector . Besides , The zero vector holds for any equation , There's no point .
3. A diagonal eigenvector is a diagonal eigenvector n Heavy , And the eigenvector is arbitrary n Order nonzero column vector .
4. If there is a matrix C Make the matrix A Contract with diagonal matrix , be A Must be a symmetric matrix . Transpose both sides of the contract equation , The transpose of diagonal matrix is equal to itself .
5. Real symmetric matrix , Must be similar to diagonal matrix , Also contract with diagonal matrix .
6. If the matrix can be similarly diagonalized , Then the rank of the matrix ＝ Number of nonzero eigenvalues .
prove ： There are invertible matrices P Satisfy P^-1AP = Diagonal matrix .
r(A) = r(P^-1AP) = r( Diagonal matrix ) = Number of nonzero eigenvalues .
7. Necessary and sufficient conditions for matrix equivalence ： Homomorphic matrix with equal rank
8. The invertible matrix must be a square matrix , Otherwise, how can it turn into E？
9. about r(AB), If A and B Full rank n Square matrix , that r(AB) It's also n.
If A It's full rank ( Yes B Do line transformation ), So the result ＝r(B). Empathy , If B Is the row full rank , result =R(A).
10. A matrix , Eigenvalues are unique , Eigenvectors are not unique . Eigenvalues are unique and well understood , Because the characteristic polynomial equation is unique , Its solution is unique . Eigenvectors are not unique because of the same eigenvalue λ Of the eigenvector of 0 Linear combination is still λ Eigenvector of . At ordinary times, what we find in solving problems is the basic solution system , Fallibility is understood as the uniqueness of eigenvectors .