The eigenvalue and eigenvector problem of 1. matrix: This kind of problem needs to solve the eigenvalue and corresponding eigenvector of a matrix, and usually needs diagonalization or similar transformation of the matrix. In the calculation process, it may involve complex matrix operation and determinant expansion, which is difficult for beginners.
2. Solution of linear equations: The solution of linear equations can be solved by gauss elimination's and Clem's rules, but in some special cases, these methods may not be directly applied and need to be deformed or introduced with auxiliary variables. In this case, the thinking of solving problems is more complicated, and the knowledge of linear algebra needs to be applied flexibly.
3. Rank and linear correlation of matrix: Rank of matrix is an important concept, which represents the number of the largest linearly independent groups of row or column vectors in matrix. Solving the rank of matrix usually requires row transformation or column transformation, but in some cases, the rank of matrix may not be easy to calculate directly. In addition, it is also a common problem to judge whether a group of vectors are linearly related, which needs to be judged by the relationship between the rank of vector group and the rank of matrix.
4. Inverse matrix and determinant of matrix: For reversible matrix, its inverse matrix can be solved by adjoint matrix method or Gauss-Jordan elimination method. But in some special cases, the matrix may not have an inverse matrix, or the value of the determinant is 0. In this case, the matrix needs to be further analyzed and processed to obtain the correct results.
In a word, the difficult problems in the linear algebra of postgraduate mathematics mainly focus on the eigenvalue and eigenvector of matrix, the solution of linear equations, the correlation between rank and linearity of matrix, and the inverse and determinant of matrix. To solve these problems, we need to have a deep understanding of the basic concepts and methods of linear algebra, and have strong logical thinking and analytical ability.