Eigenvalues and eigenvectors · Definition

Eigenvalue and eigenvector

linear algebraeigenvalues

Before you start you need to know

Almost everything done with a large matrix (powers, dynamical systems, vibrations) gets simpler once we find the directions the matrix does not turn.

Definition

An eigenvector \(v\) of a square matrix \(A\) is a nonzero vector that \(A\) only stretches, shrinks or flips, without taking it off its line; the factor \(\lambda\) is its eigenvalue.

\[Av=\lambda v,\quad v\neq 0\]

A vector \(v\neq0\) is an eigenvector of \(A\) when \(Av\) is a multiple of \(v\): \(Av=\lambda v\). Geometrically, \(A\) maps the line spanned by \(v\) into itself, stretching it if \(|\lambda|>1\), shrinking it if \(|\lambda|<1\) and flipping it if \(\lambda<0\). For instance, for \(A=\begin{pmatrix}4&1\\2&3\end{pmatrix}\), \(A(1,1)=(5,5)\) and \(A(1,-2)=(2,-4)=2(1,-2)\): the eigenvalues are 5 and 2.

We require \(v\neq0\) because \(A0=\lambda0\) holds for every \(\lambda\) and would tell us nothing. By contrast, \(\lambda=0\) can be an eigenvalue: this happens exactly when \(Av=0\) has nonzero solutions, that is, when \(\det A=0\). If \(v\) is an eigenvector, so is any nonzero multiple, with the same \(\lambda\), and \(A^kv=\lambda^kv\).

An eigenvalue need not be real. The 90° rotation of the plane, \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\), fixes no line and its eigenvalues are \(\pm i\). When an \(n\times n\) matrix has \(n\) independent eigenvectors, they form a basis in which the matrix is diagonal, an idea developed in diagonalization.

Typical mistakes: accepting the zero vector as an eigenvector, believing each eigenvalue has a single eigenvector (there is a whole line, or subspace, of them) and thinking that the sum of eigenvectors for different eigenvalues is another eigenvector. How to find eigenvalues without guessing is the subject of the characteristic polynomial.

Think about it

Which eigenvectors does the identity matrix have?

Every nonzero vector is an eigenvector, with eigenvalue 1, because \(Iv=v\).

If \(v\) is an eigenvector of \(A\) with eigenvalue \(\lambda\), which eigenvalue corresponds to it for \(3A\)?

\(3\lambda\), since \((3A)v=3(\lambda v)=(3\lambda)v\).

Exercises 10

  1. Let \(A=\begin{pmatrix}4&1\\2&3\end{pmatrix}\). Which of these numbers is an eigenvalue of \(A\)?
  2. Let \(A=\begin{pmatrix}1&2\\2&1\end{pmatrix}\) and \(v=(1,1)\). What is the eigenvalue associated with \(v\)?
  3. The vectors \((1,0)\) and \((1,-1)\) are eigenvectors of \(A=\begin{pmatrix}3&1\\0&2\end{pmatrix}\). What are their eigenvalues, in that order?
  4. Let \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\). Which of these vectors is an eigenvector of \(A\)?
  5. Can the zero vector be an eigenvector?
  6. If \(Av=4v\) with \(v\neq0\), what is \(A^2v\)?
  7. How many real eigenvalues does the 90° rotation \(A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) have?
  8. If \(v\) is an eigenvector of \(A\) with eigenvalue 2, what is \(A(-3v)\)?
  9. If 0 is an eigenvalue of the square matrix \(A\), what follows?
  10. A \(3\times3\) matrix has eigenvectors \(v_1,v_2,v_3\) with eigenvalues 2, 3 and 5. What is \(A(v_1+v_2)\)?

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §6.1
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §5.A
  • David C. Lay, Steven R. Lay, Judi J. McDonald, Linear Algebra and Its Applications, 6th edition, Pearson, 2021. §5.1
  • Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra, 4th edition, Pearson / Prentice Hall, 2003. §5.1