Beginner matrix questions, how to solve matrix problems

a = where is the eigenvalue, which is the corresponding non-zero eigenvector (this is the definition of eigenvalue and eigenvector).

By a2=a, then a2=

And a 2 = a*(a) = a* = a = * = 2 =

Get 2 = => =0 or 1

There is a problem with the upstairs friend's solution, i.e. a(a-i)=0 cannot push out a=0 or a=i, because 0 here is a 0 matrix.

Imagine a diagonal matrix, as long as the numbers on the diagonal are only 0 and 1, they all satisfy a(a-i)=0

tr(a) is not necessarily equal to 1, because as I have already said, a can be a diagonal matrix, as long as there are only 0 and 1 on the diagonal, as is: a is.

satisfying a 2 = a, but tr(a) = 3, in fact, tr(a) = r(a), that is, the trace of a is equal to the rank of a.

According to the definition of eigenvalue, if there is ax jx, j is an eigenvalue of a, and x is the eigenvector corresponding to the eigenvalue j. However, there is a better solution to this problem.

Because a 2=a

So a 2-a = 0

a(a-1)=0

a(a-i)=0 (i is the identity matrix).

It follows that a 0 or a i

When a=0 (zero matrix), the eigenvalue is 0

Because the square of 0 is equal to 0, and the square of 1 is equal to 1.

The number of lines has long forgotten the light, and it can't help you.

Solution: Consists of a 2-3a+4e=0

Get a(a-3e) =4e

Therefore, a can be reversed, and a -1 = 1 4)(a-3e) is then replaced by a 2-3a+4e=0

A(A+4E)-7(A+4E) +32E = 0, so (A-7E)(A+4E) =32E

So a+4e is reversible, and (a+4e) -1 = 1 coarse 32)(a-7e).

In mathematics, a matrix is a set of complex or real numbers arranged in a rectangular array, which originally came from a square matrix composed of coefficients and constants of a system of equations. This concept was first proposed by the 19th-century British mathematician John Kelly.

Matrices are a common tool in advanced algebra and are also commonly found in applied mathematics disciplines such as statistical analysis. In physics, matrices have applications in circuits, mechanics, optics, and quantum physics; In computer science, 3D animation also requires the use of matrices. The operation of matrices is an important problem in the field of numerical analysis.

Decomposing matrices into combinations of simple matrices can simplify the operation of matrices in theory and practical applications. For some matrices with wide applications and special forms, such as sparse matrices and quasi-diagonal matrices, there are specific fast operation algorithms. For the development and application of matrix-related theory, please refer to Matrix Theory.

In the fields of astrophysics, quantum mechanics, etc., infinite-dimensional matrices will also appear, which is a kind of generalization of matrices.

The main branch of numerical analysis is devoted to the development of efficient algorithms for matrix computation, a topic that has been a topic for centuries and an expanding field of study. The matrix factorization method simplifies both theoretical and practical calculations. Algorithms tailored to specific matrix structures, such as sparse and near-angle matrices, speed up calculations in finite element methods and other calculations.

Infinite matrices occur in the theory of planets and atoms. A simple example of an infinite matrix is a matrix representing a derivative operator of a function's Taylor series.

I hope I can help you with your doubts.

Directly substitute am=ma, a b ) = ( a+2b c+2d )

2a-c 2b-d ) b -d ), so b=0, c+2d=2a-c=0, d=-a, c=2a, so.

m=a ( 2 -1) =aa。

A matrix that is not a square matrix can also have an "inverse". A more typical "inverse" is the moore-penrose inverse (moore-penrose pseudoinverse, the reason why there is a pseudo, I think, is because it is not fully compatible with the inverse of the square, and it also exists when the square itself is irreversible). If matrix A is m*n, then its moore-penrose inverse b is n*m, satisfying:

1)aba=a；

2)bab=b；

3)(ab)^*= ab；where a* refers to the conjugate transposition of a, 4) (ba) * = ba. For real matrices, conjugate transpose is transpose.

It is easy to see that if A is a phalanx, then the Moore-Penrose inverse of A is the only one, which is its inverse. Even if A is not a phalanx, its moore-penrose inverse is unique. In addition, if you take the moore-penrose inverse twice, then it will change back to the original matrix, that is, the inverse of a is a.

For more information, you can search the wiki for moore-penrose pseudoinverse, I would have liked to put the link in the reference section, but I didn't. Some algebra textbooks (such as Qiu Weisheng's "Advanced Algebra", I have seen the first edition, which involves more things around algebra) also have some content (or exercises) that can be referred to.

If you multiply the middle two of the four terms separately, it is equal to k, and the coefficient can be advanced by the surface.