Eigenvalue and eigenvector are probably one of the most important concepts in linear algebra. Who can expect a simple equation like Av = λv is so significant? From machine learning, quantum computing, and physic, many mathematical and engineering problems can be solved by finding the eigenvalue and eigenvectors of a matrix. Let’s not only discover what it is but also answer why it is so important. We will look into the Google PageRank to see how page ranking works.
By definition, scalar λ and vector v are the eigenvalue and eigenvector of A if
Visually, Av lies along the same line as the eigenvector v.
Here are some examples.
However, Ax does not usually equal to λx. Only some exceptional vectors satisfy the condition. If the eigenvalue is greater than one, the corresponding Avᵢ will expand. If it is smaller than one, it will shrink.
But before getting into details, let’s pause and appreciate the beauty of such an abstract concept first. Many problems can be modeled with linear transformations with solutions derived from eigenvalues and eigenvectors. Let’s detail it with an abstract example first before real problems with a billion-dollar idea — Google’s PageRank. In many systems, we can express the properties in a vector with their rates of change linearly depend on the current properties (e.g. the population growth rate depends on the current population and GDP linearly.). The general equation is
So let’s take a guess on u(t) that satisfies the equation above. Since the derivative of an exponential function equals itself, we start with an exponential function of t and multiply it with a vector x — the output will be a vector.
From the calculation above, our solution for u(t) is
Next, we will find its complete solution. Our first order derivative equation is a linear function.
For linear functions, the complete solution is the linear combination of particular solutions. If u and v are the solutions, C₁u + C₂v is also the solution. From our previous example with eigenvalues λ = 4, -2 and -2, the complete solution will be
If a system will reach a stable state, then all eigenvalues have to be negative. At time t=0, we can measure the initial state u(0), say [u₀₁, u₀₂, u₀₃]ᵀ, and solve the constant C₁, C₂, and C₃.
Let’s illustrate the idea with the harmonic oscillator. We pick this example because the harmonic oscillator and its close cousin, the quantum harmonic oscillator, are almost everywhere in studying particle physics, quantum mechanics, string theory or physics in general. Let’s start with the famous F=ma equation and use eigenvalues and eigenvectors to solve a second-order derivative. Since we do have the freedom to pick the unit for mass, physicists often set m=1 to simplify the discussion, i.e.
We reformulate the harmonic oscillator problem into a matrix form in the bottom right below.
This has the same form as our last example and therefore, we can use the eigenvalues and eigenvectors of A to form the complete solution.
This is not an isolated example in demonstrating the power of eigenvalues. Nature seems to have an eigenvector cookbook when making its design. The famous time-independent Schrödinger equation is expressed with eigenvalues and eigenvectors. All observed properties are modeled by eigenvalues in quantum mechanics. They are many other examples including machine learning and one of the biggest eigenvector computed, Google PageRank.
Fundamentally, many systems can be modeled as
Let’s study the time sequence model a little more for the purpose of machine learning.
First, we assume the initial state u₀ to be an eigenvector of A. Therefore, the future states can be computed as
In short, we can simplify the calculation by replacing the power of a matrix (Aᵏ) with the power of a scalar. Next, consider A has n linearly independent eigenvectors which form a basis of Rⁿ. We can decompose any vector of Rⁿ into this basis and simplify the calculation by computing the power of the eigenvalue again.
If a system will reach a stable state, we should expect λᵢ to be smaller or equal to 1. To compute the stable state, we can ignore terms with λᵢ smaller than 1 and just find the eigenvector associated with λᵢ = 1.
Let’s discuss a real multi-billion idea to realize its full potential. Let’s simplify the discussion which assumes the whole internet contains only three web pages. The element Aᵢⱼ of a matrix A is the probability of a user going to page i when the user is on page j.
If we sum up all the possibilities of the next page given a specific page, it equals 1. Therefore, all columns of A sum up to 1.0 and this kind of matrix is called the stochastic matrix, transition matrix or Markov matrix.
Markov matrix has some important properties. The result of Ax or Aᵏx always sums up to one with its columns. This result indicates the chance of being on page 1, 2 and 3 respectively after each click. So it is obvious that it should sum up to one.
Any Markov matrix A has an eigenvalue of 1 and other eigenvalues, positive or negative, will have their absolute values smaller than one. This behavior is very important. In our example,
For a Markov matrix, we can choose the eigenvector for λ=1 to have elements sum up to 1.0. Vectors v with elements sum up to one can be decomposed using the eigenvectors of A with c₁equals to 1 below.
Since u₁, u₂, …, and un are eigenvectors, Aᵏ can be replaced by λᵏ. Except for eigenvalue λ=1, the power of the eigenvalue (λᵏ) for a Markov matrix will diminish, as the absolute values of these eigenvalues are smaller than one. So the system reaches a steady state that approaches the eigenvector u₁ regardless of the initial state. And both Aᵏ and the steady state can be derived from the eigenvector u₁ as below.
In our example, the chance we land on page 1, 2 and 3 are about 0.41, 0.34 and 0.44 respectively. This concept has many potential applications. For instance, many problems can be modeled with Markov processes and a Markov/transition matrix.
I am exaggerating when I say this is a billion-dollar idea. But the PageRank algorithm named after Google co-founder Larry Page has a similar concept. It is the first Google search ranking algorithm even it is heavily modified now with added ranking algorithms to improve user experience and to avoid manipulation.
The core idea can be conceptualized as the following. PageRank outputs a probability distribution of pages you may land on after a random walk, by following web links in a Web page. This probability acts as the ranking of a Web page. When many Web pages link to your Web page, Google will rank it higher considering it as a good indicator of popularity.
In our previous example, we have not discussed how to derive the values in the Markov matrix. Conceptually, we compute a page rank that equals the sum of other page ranks linked to this page divided by its total number of outbound pages.
Mathematically, PageRank tries to solve the PageRank R (an eigenvector) in the following equation. It initializes R with equal probability at the beginning and performs the calculation iteratively until it reaches a steady state.
lᵢⱼ is the ratio between the number of the outbound page from page j to page i to the total number of the outbound page of j. Each column of l adds up to 1 and it is a Markov matrix. This equation has an enormous similarity with the example we discussed before if we ignore the damping factor d. This factor is introduced because the random walk does not take forever.
For Google, they do not compute the eigenvectors directly. In our previous example, the power of A converges very fast which the column of A³ converge to eigenvector u₁ already.
The PageRank paper has demonstrated that with 322 million page links, the solution converges to a tolerable limit in 52 iterations. So the solution scales unexpectedly well.
The Markov matrix leads us to the equation below which the steady state depends on one principal component.
In machine learning, information is tangled in raw data. Intelligence is based on the ability to extract the principal components of information inside a stack of hay. Mathematically, eigenvalues and eigenvectors provide a way to identify them. Eigenvectors identify the components and eigenvalues quantify its significance. As a preview, the equation below decomposes information in A into components. We can prioritize them based on the square root of eigenvalues and ignore terms with small α values. This reduces the noise and helps us to extract the core information in A. (Details on a later article.)
Hope you can see the beauty for Ax = λx for now. Eigenvalues and eigenvectors can be calculated by solving (A - λI) v = 0. To have a solution other than v=0 for Ax = λx, the matrix (A - λI) cannot be invertible. i.e. it is singular. i.e. its determinant is zero. det(A - λI) = 0 is called the characteristic polynomial. The eigenvalues are the root of this polynomial.
The eigenvalues are:
Apply Av = λv, we solve:
Let’s detail the step with a more complicated example,
To find the eigenvalue λ,
The possible factors for 16 are 1, 2, 4, 8, 16.
Let’s calculate the eigenvector for eigenvalue λ = 4 through row reduction.
We have three variables with 2 equations. We set x₃ arbitrary to 1 and compute the other two variables. So for λ=4, the eigenvector is:
We repeat the calculation for λ=-2 and get
With 3 variables and 1 equation, we have 2 degrees of freedom in our solution. Let’s set 1 to one degree of freedom one at a time with the other(s) to 0. i.e. setting x₂=1, x₃=0, and x₂=0, x₃=1 separately, the calculated eigenvectors are:
Note that the solution set for eigenvalues and eigenvectors are not unique. we can rescale the eigenvectors. We can also set different values for x₂, x₃ above. Hene, it is possible and desirable to choose our eigenvectors to meet certain conditions. For example, for a symmetric matrix, it is always possible to choose the eigenvectors to have unit length and orthogonal to each other.
In our example, we have a repeated eigenvalue “-2”. It generates two different eigenvectors. However, this is not always the cases — there are cases where repeated eigenvalues do not have more than one eigenvector.
Let’s assume a matrix A has two eigenvalues and eigenvectors.
We can concatenate them together and rewrite the equations in the matrix form.
We can generalize it into any number of eigenvectors as
where V concatenates all the eigenvectors and Λ (the capital letter for λ) is the diagonal matrix containing the eigenvalues.
A square matrix A is diagonalizable if we can convert it into a diagonal matrix like,
An n × n square matrix is diagonalizable if it has n linearly independent eigenvectors. If a matrix is symmetric, it is diagonalizable. If a matrix does not have repeated eigenvalue, it always generates enough linearly independent eigenvectors to diagonalize a vector. If it has repeated eigenvalues, there is no guarantee we have enough eigenvectors. Some will not be diagonalizable.
If A is a square matrix with N linearly independent eigenvectors (v₁, v₂, … & vn and corresponding eigenvalues λ₁, λ₂, … & λn), we can rearrange
However, there are some serious limitations on eigendecomposition. First, A needs to be a square matrix. Second, V may not be invertible. As a preview, SVD solves both problems and SVD decomposes a matrix into orthogonal matrices which are easier to manipulate. But for matrices like symmetric matrices, this is not a problem. So eigendecomposition still has its role.
The power of a diagonalizable matrix A is easy to compute.
This demonstrates the power of decomposing a matrix for easier manipulation in solving problems like the time sequence model discussed before.
Properties of eigenvalue & eigenvectors
- Ax lies on the same line as the eigenvector x (same or opposite direction).
- The sum of eigenvalues equals the trace of a matrix (sum of diagonal elements).
- The product of eigenvalues equals the determinant.
- Both conditions above serve as a good insanity check on the calculations of eigenvalues.
- If no eigenvalue is repeated, all eigenvectors are linearly independent. Such an n × n matrix will have n eigenvalues and n linearly independent eigenvectors.
- If eigenvalues are repeated, we may or may not have all n linearly independent eigenvectors to diagonalize a square matrix.
- The number of positive eigenvalues equals the number of positive pivots.
- For Ax = λx,
- If A is singular, it has an eigenvalue of 0. An invertible matrix has all eigenvalues non-zero.
- Eigenvalues and eigenvectors can be complex numbers.
- Projection matrices always have eigenvalues of 1 and 0 only. Reflection matrices have eigenvalues of 1 and -1.
Eigenvalues quantify the importance of information along the line of eigenvectors. Equipped with this information, we know what part of the information can be ignored and how to compress information (SVD, Dimension reduction & PCA). It also helps us to extract features in developing machine learning models. Sometimes, it makes the model easier to train because of the reduction of tangled information. It also serves the purpose to visualize tangled raw data. Other applications include the recommendation systems or financial risk analysis. For example, we suggest movies based on your personal viewing behavior and others. We can also use eigenvectors to understand the correlations among data. Develop trends of the information and cluster information to find the common factors, like the combination of genes that triggers certain kind of disease. And all of them start from the simple equation:
Matrices are not created equally. It is important to know the properties of some common matrices in machine learning. Next, we will take a look at it.