The relationship between matrix equivalence, contract and similarity;
Equivalence (only the same level)-> Contract (same as positive and negative inertia index)-> Similarity (rank, positive and negative inertia index, same characteristics) deepens the intimate relationship of matrix step by step. Similar matrix must be equivalent matrix, but equivalent matrix is not necessarily similar matrix. The equivalent matrix of P=E is a similar matrix.
The contract matrix must be an equivalent matrix, and the equivalent matrix is not necessarily an equivalent matrix with the same positive inertia index. The contract matrix is not necessarily a similar matrix. The similarity matrix is not necessarily a contract matrix, and the orthogonal contract matrix must be a similarity matrix. If both A and B are real symmetric matrices of order n and have the same characteristic roots, then both A and B are similarly contracted.
How to learn math well?
Listen carefully in class, process and sort out what the teacher said (generally, children with poor math don't have time to take notes, and good students don't have to take notes), and translate what the teacher said into natural language. For example, I say that two vector lines are equivalent to b=λa, which translates into a multiple relationship between b and a. This is a natural language, which is easy to understand and deepen understanding.
Using graphic memory, Buzan's mind map (it's actually a bit messy to do mind maps in high school mathematics) tells us that graphics can easily help memory (improve memory ability 100 times or more), so I always say to look at pictures and speak in class, and use graphics to help remember formulas and help solve problems.
Correct after class, and correct any mistakes. Philosophically speaking, it is impossible for people to step into the same river twice, but students who make mistakes in their studies often make mistakes, and preventing them from making mistakes again can greatly improve their grades.
Understand the topic, why do you want to do it? Students don't have time to learn how to solve problems by Paulia, but they are guided by appropriate question strings, such as asking themselves, why do you want to do this step, why do you want to simplify it, why do you want to ... This kind of thinking habit can improve your problem-solving ability, and I also explain it in class.
Proper practice, sea tactics, I don't recommend it, but it is the most effective, but it is not difficult to practice on the basis of the above. Maybe many teachers can't get home. Let's talk about it.