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Unit 7: Eigenvalues, Eigenvectors, and Dynamical Systems

The vectors a matrix refuses to knock off their own line, and how they predict everything a system does over time.

12 topics · 42 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

What an Eigenvector Is

  1. How it worksHow does Av equal lambda v capture a vector that only gets stretched, never rotated, by a matrix?
  2. Worked exampleShow that (1, 1) is an eigenvector of the matrix [[2, 1], [1, 2]] and find its eigenvalue.
  3. Common mistakeWhy isn't an eigenvector unique — any scalar multiple of it works just as well?
  4. Common mistakeWhy doesn't every vector a matrix happens to stretch count as one of its eigenvectors?
  5. Going furtherWhat happens to an eigenvector's arrow once its eigenvalue turns out to be negative?
Your turn1 of 2
Show that (1, 1) is an eigenvector of the matrix [[3, 2], [2, 3]] and find its eigenvalue.

The Characteristic Polynomial

  1. How it worksHow does setting the determinant of A minus lambda I to zero produce the characteristic polynomial?
  2. Worked exampleFind the characteristic polynomial of the matrix [[2, 1], [1, 2]].
  3. Common mistakeWhy doesn't the characteristic polynomial's constant term have nothing to do with the determinant of A?
Your turn1 of 2
Find the constant term of the characteristic polynomial of the matrix [[3, 2], [7, -4]] — this is det(A).

Finding Eigenvalues and Eigenvectors

  1. How it worksHow does plugging each eigenvalue back into A minus lambda I locate its eigenvectors inside the null space?
  2. Worked exampleFind the eigenvalues and eigenvectors of the matrix [[4, 1], [2, 3]].
  3. Worked exampleFind the eigenvalues and eigenvectors of the second matrix [[0, 1], [-2, -3]], which has both eigenvalues negative.
  4. Common mistakeWhy doesn't a repeated eigenvalue mean there's only ever one independent eigenvector attached to it?
Your turn1 of 2
Find the SMALLER eigenvalue of the symmetric matrix [[2, 4], [4, 5]].

The Trace and Determinant Shortcut for Eigenvalues

  1. How it worksHow does the trace giving their sum and the determinant giving their product pin down both eigenvalues of a 2 by 2 matrix?
  2. Worked exampleFind the eigenvalues of the matrix [[5, 2], [2, 2]] using its trace and determinant.
  3. Ties togetherHow does this shortcut connect the trace and determinant back to the coefficients of the characteristic polynomial?
Your turn1 of 2
Using the trace-and-determinant shortcut, find the LARGER eigenvalue of the matrix [[2, 4], [4, 5]].

Eigenvalues of Triangular and Diagonal Matrices

  1. How it worksHow does the determinant of a triangular matrix minus lambda I factor immediately into its diagonal entries?
  2. Worked exampleFind the eigenvalues of the matrix [[3, 5, 7], [0, 2, 4], [0, 0, 6]].
  3. Common mistakeWhy doesn't finding the eigenvalues of a general matrix ever come this easily, the way it does for a triangular one?
Your turn1 of 2
Find the third eigenvalue of the triangular matrix [[4, 3, 8], [0, -4, -8], [0, 0, -5]].

Diagonalizing a Matrix

  1. How it worksHow does packing eigenvectors into S and eigenvalues into Lambda let you write A as S Lambda S inverse?
  2. Worked exampleFind the diagonalization of the matrix [[4, 1], [2, 3]] by finding S and Lambda so that A equals S Lambda S inverse.
  3. Common mistakeWhy can't every matrix be diagonalized, even a matrix with no zero eigenvalues at all?
  4. Going furtherWhat happens to A squared and A cubed once you have A written as S Lambda S inverse?
Your turn1 of 2
A matrix is diagonalizable as A = S Lambda S inverse, where one of its eigenvalues is 2. Find the corresponding eigenvalue of A cubed.

Powers of a Matrix via Diagonalization

  1. How it worksHow does A to the k power collapse down to S times Lambda to the k power times S inverse?
  2. Worked exampleFind A to the 10th power for the matrix [[2, 0], [0, -1]] using its eigenvalues.
  3. Common mistakeWhy doesn't computing A to a high power by direct repeated multiplication work just as well as diagonalizing first?
Your turn1 of 2
A diagonalizable matrix has an eigenvalue equal to 2. Find the corresponding eigenvalue of A to the 5 power.

Similar Matrices

  1. How it worksHow does B equal S inverse A S mean A and B describe the same map, just viewed in two different bases?
  2. Worked exampleFind B equal to S inverse A S for A equal to [[2, 0], [0, 5]] and S equal to [[1, 1], [0, 1]], and compare their eigenvalues.
  3. Common mistakeWhy doesn't similarity between two matrices mean they have the same entries, just rearranged?
  4. Going furtherWhat single property of a matrix stays exactly the same across every matrix similar to it?
Your turn1 of 2
For A = [[3, 0], [0, 2]] and S = [[1, 5], [0, 1]], compute B = S inverse A S. Find the (1,2) entry of B.

Difference Equations and Eigenvalues

  1. How it worksHow does writing the starting vector as a combination of eigenvectors let you predict any future step instantly?
  2. Worked exampleFind a formula for the Fibonacci numbers using the eigenvalues of the matrix [[1, 1], [1, 0]].
  3. Common mistakeWhy doesn't the largest eigenvalue alone tell you the whole long-run story once it turns out negative?
  4. Going furtherWhat happens to a difference equation's long-run behavior once the largest eigenvalue has absolute value less than 1?
Your turn1 of 2
Find the difference between the two eigenvalues of the matrix [[2, 4], [4, 5]] (larger minus smaller).

Differential Equations and Eigenvalues

  1. How it worksHow does substituting v times e to the lambda t into dx/dt = Ax turn the problem into an eigenvalue problem?
  2. Worked exampleSolve the system dx/dt = 2x, dy/dt = 3y given x(0) = 1 and y(0) = 1.
  3. Common mistakeWhy doesn't a negative eigenvalue in a differential equation system mean the solution itself goes negative?
Your turn1 of 2
Solve the decoupled system dx/dt = 1x, dy/dt = 1y with x(0) = 6, y(0) = 2. Find x at time t = 0.31.

Complex Eigenvalues and Rotation

  1. How it worksHow do complex eigenvalues always show up in conjugate pairs for a matrix with real entries?
  2. Worked exampleFind the eigenvalues of the rotation matrix [[0, -1], [1, 0]].
  3. Going furtherWhat happens when you try to find a rotation matrix's eigenvectors using only real numbers?
Your turn1 of 2
The matrix [[3, -6], [6, 3]] has complex eigenvalues a plus or minus bi. Find the magnitude (absolute value) of these eigenvalues.

Repeated Eigenvalues and Defective Matrices

  1. How it worksHow does a repeated eigenvalue sometimes produce only one independent eigenvector instead of two?
  2. Worked exampleShow that the matrix [[2, 1], [0, 2]] has only one independent eigenvector for its repeated eigenvalue.
  3. Common mistakeWhy doesn't an eigenvalue repeated twice guarantee that two independent eigenvectors go along with it?
Your turn1 of 2
Find the repeated eigenvalue of the matrix [[4, 3], [0, 4]].