Article from WeChat public account:Quantum (ID:QbitAI), author: thirteen, fish, sheep, drawing from the title: vision China

In the early morning of August, the mathematic genius and Fields Prize winner Tao Zhexuan opened an email from three unfamiliar physicists.

Three people explained in the email:

We stumbled upon a formula that, if it was correct, would create an unexpected relationship between some of the most basic and important objects in linear algebra.

However, Tao Zhexuan’s first reaction was:

So short, simple things should have appeared in textbooks long ago. This can’t be true.

In fact, Tao Zhexuan has never liked to be consulted in this way, and even wrote a warning on his homepage: Don’t bother me with your manuscript.

But the three physicists were surprised that they received a reply from Tao Zhexuan just two hours later.

What is even more unexpected is that after a week and a half, they also published a paper together explaining the proof process of this formula.

What kind of formula is so popular with Tao Zhexuan?

Solve the feature vector.

Yes, this is the basic mathematical solution formula.

According to traditional solutions:

Calculate feature polynomial→solve eigenvalues→solve homogeneous linear equations to derive feature vectors.

These three physicists have unexpectedly discovered another wonderful solution in the process of studying “neutrinos”:

Knowing the eigenvalues, you only need to list a simple equation, and the eigenvectors can be solved.

Three physicists. From left to right: Zhang Xining, Peter Denton and Stephen Parke.

As Tao Zhexuan said:

This formula looks incredibly good.

I have never thought about it. The eigenvalues ​​of the submatrices encode the hidden information of the original matrix eigenvectors.

Yale University mathematician Van Vu uses the words “amazing” and “fun” to describe the discovery.

A Hacker News netizen even believes that the theoretical value of this formula is above Clem’s Law.

Note: Clem’s law is the basic theorem in linear algebra, and the solution of the n-ary equations is calculated by the determinant.

How come the new method?

First, let’s review the feature vectors and eigenvalues ​​we know well.

Multiplying a matrix by a vector is equivalent to doing a linear transformation. But the direction of this vector tends to change.

But if there is a matrix A, let the vector v remain linear after the linear transformation, just stretch or compress a certain multiple, namely: Av = λv.

Then, this vector v is the feature vector, and λ is the feature value.

In the current textbooks, it is easier to obtain feature values ​​from known feature vectors, but it is more convenient to find the feature values ​​of the matrix than to obtain feature vectors.

But three physicists found that when calculating the probability of neutrino oscillation:

The geometric nature of feature vectors and eigenvalues ​​is actually the rotation and scaling of space vectors. The neutrino’s three flavors (electronic, mu, τ), is not equivalent to the space between the three vectors Transform?

The neutrino oscillation is a quantum mechanical phenomenon. It is found that the three neutrinos, the electron neutrino, the muon neutrino and the muon neutrino, can be transformed into each other, and this is the phenomenon of neutrino oscillation.

Source: Quantamagazine

Physicists realize that there may be more general patterns between eigenvectors and eigenvalues. Thus, the veil of the new formula was unveiled.

Create a submatrix by deleting the rows and columns of the original matrix.

The sub-matrix and the eigenvalues ​​of the original matrix are combined to calculate the eigenvector of the original matrix.

In short, the eigenvalues ​​are known, and an equation can be used to find the eigenvectors.

Source: Quantamagazine

How many new formulas does this have?

Mathematical genius, Fields Award winner Tao Zhexuan commented:

The extraordinary thing about the new formula is that in any case, you don’t need to know any elements in the matrix to calculate anything you want.

certification process

In Tao Zhexuan’s reply, he also attached three proofs of this new formula, and later published papers with three physicists Peter Denton, Stephen Parke, and Zhang Xining.

First define A as an n x n Hermitian matrix with eigenvectors λi(A) and normative eigenvectors vi.

Hermit Matrix (Hermitian Matrix) can convert feature vectors into real numbers, which is more suitable for solving real-world problems.

Each element in the feature vector is labeled vi,j.

By deleting the jth line and the jth column, you can get the submatrix Mj of A with a size of (n-1) x (n-1), whose characteristic value is λk(Mj).

First, by proof, you can get a Cauchy-Bine (Cauchy-Binet) type formula.

Lemma 1. Let one of AThe eigenvalue is 0, and without loss of generality, λn(A)=0. Then for any matrix B of size n x (n-1), we can get:

Next, you can enter the derivation of the new formula.

Lemma 2. The norm square of each element of the eigenvector is related to its eigenvalue and submatrix eigenvalue.

Then it can be proved that let j = 1 and i = n. Shifting A by λn(A)In, such that λn(A)=0; this also transforms A And all remaining eigenvalues ​​in Mj, so Equation 2 becomes:

Note that the right side of Equation 3 is det(M1).

Next, Lemma 1 is applied in B=(0, In-1). We find that the left side of Formula 1 is

The right side of Equation 1 is det(M1).

Proof: For any λ that is not the eigenvalue of A,

For, j∈[1,n],

Further simplification, and take the limit λ→λi(A),

The diagonal element to the right of Equation 7 provides the left half of Equation 2. By the definition of conjugate, the diagonal element to the left of Equation 7 determines the submatrix of λi(A)In-A.

Application Lemma 2, the inevitable conclusion is that if an element in the feature vector disappears, vi,j=0, then the eigenvector equation of matrix A will be transformed into its submatrix M A eigenvector equation for j.

The impact of this discovery

In short, this latest achievement by physicists will allow one to calculate feature vectors using only eigenvalue information.

In the current textbooks, it is easier to obtain feature values ​​from known feature vectors, but it is more convenient to find the feature values ​​of the matrix than to obtain feature vectors.

In other words, this achievement reveals the new facts of basic mathematics.

More importantly, in the real world, whether in mathematics, physics or engineering, many problems involve the calculation of eigenvectors and eigenvalues.

For example, calculate the probability of neutrino oscillation.

For example, in the field of machine learning, data dimensionality reduction, face recognition, all involve the practical application of matrix eigenvalue/feature vector theory.

John Beacom, a particle physicist at Ohio State University, points out that this theory has broad application prospects and will even open the door to a new world.

Co-authorship of physicists and math genius

Invited by three physicists, the mathematician who proved the new formula is a recognized mathematical genius Tao Zhexuan

Tao Zhexuan

He was a high school student at the age of 7 and went to college at the age of 9. At the age of 13, he won the gold medal in the International Olympic Mathematical Competition. He is the holder of the youngest winner of the IMO Gold and Silver Bronze.

At the age of 24, he became a tenured professor of mathematics at UCLA. At the age of 31, he won the Fields Medal, which is known as the Nobel Prize in Mathematics, and became the second Chinese mathematician to receive this honor.

And three physicists, one is the assistant physicist Peter Denton in the Brookhaven National Laboratory in the United States (Peter B.Denton) . In 2016, he graduated from the Physics Department of Vanderbilt University.

The other is New Zealand physicist Stephen Parker (Stephen J. Parke). He is a Distinguished Scientist and Director of the Department of Theoretical Physics at the Fermi National Accelerator Laboratory in the United States, focusing on neutrino physics and top quark physics.

The last author, Zhang Xining (Xining Zhang) is also a Chinese face, studying at the University of Chicago, engaged in theoretical particle physics research, is Stephen · Parker’s disciple.

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