How To Calculate Eigenvalues

The concept of ​eigenvalues​ is obscure but comes in very handy for mathematicians and physical scientists faced with certain interesting problems.

To understand an eigenvalue, imagine having a function (e.g., ​y​ = ​x​² + 6​x​, or ​y​ = log 4​x​) that you could put through some process such that the result would be the same as multiplying the whole function by a constant value. Such a function would qualify as an ​eigenfunction​, and the constant would be an eigenvalue.

•"Eigen" is German for "same."

To best understand eigenvalues and eigenfunctions, and be able to calculate eigenvalues yourself, you need a basic understanding of matrices. These mathematical tricks are used to determine say, the bond order of NO₂ (nitrogen dioxide) and other molecules, because electron behavior in atoms is determined by wavefunctions that qualify as eigenfunctions.

A matrix is an array of numbers ordered in rows and columns, which may number from 1 to ​n​. The dimensions of matrices are given as row-by-column; for example, the following is a 2-by-3 matrix:

\(\begin{bmatrix}
3 & 0 & 4 \
1 & 3 & 5 \
\end{bmatrix}\)

Matrices can be added together if they are the same size (that is, have the same number of rows and the same number of columns). They can also be multiplied together by a stepwise process under the same conditions. In addition, any matrix can be multiplied by a vector, which is a 1-by-​n​ or ​n​-by-1 matrix; this includes other vectors.

Say you have an ​n​-by-​n​ or "square" matrix ​A​, a nonzero ​n​-by-1 vector ​v​, and a scalar ​λ​, such that the following equation is satisfied:

\(\bold{Av} = λ\bold{v}\)

Any value of ​λ​ for which this equation has a solution is known as an eigenvalue of the matrix ​A​.

Don't let your mind treat the above expressions as product. ​A​ is an ​operator​ on, or a linear transformation of, the vector ​v​, this computation being possible only because ​A​ and ​v​ both have ​n​ rows.

The derivation is complicated, but in atomic chemistry, the Hamiltonian operator "H-bar" is used to express the kinetic and potential energy of a system:

\(\hat H=−\dfrac{ℏ}{2m}∇^2+\hat V(x,y,z)\)

This is used to write a form of the ​Schrodinger wavefunction equation​ in quantum mechanics:

\(\hat Hψ(x,y,z)=Eψ(x,y,z)\)

Here ​E​ represents the eigenvalues satisfying this equation.

From the equation Av = λv, you get ​A​ ​v​ − λ​v​ =0. This leads to:

\(\bold{A v} − λ(\bold{I v})=0\)

Where ​I​ is the 2-by-2 identity matrix with rows of [​λ​ 0] and [0 ​λ​], leading to 1 when multiplied by the scalar ​λ​. This result yields:

\((\bold{A} – λ\bold{I})\bold{v} = 0\)

Which if ​v​ is nonzero, only has a solution if the absolute value of ​A​− ​λ​​I​, or |​A​ − ​λ​​I​|, is zero. If you do these by hand, it involves solving a quadratic equation and can be tedious.

To multiply two matrices together, for each point in the product matrix, you multiply the corresponding points together and add this to the products of the remaining row and column elements in the row and column to which the new point belongs.

In multiplying two 2-by-2 matrices ​A​ and ​B​ together, if the first row of ​A​ is [1 3] and the first column of ​B​ is [2 5], the number in the first column and row of the new matrix would be [(1 × 2) +(3 × 5)] = 15, and correspondingly for the other three points.

In the Resources, you'll find a matrix calculation tool that allows you to find eigenvalues and more for a matrix of almost any conceivable size.