Used in AI HL for long-run behaviour of systems (e.g. coupled populations, Markov chains). The GDC computes them directly - no characteristic polynomial by hand.
In short: Eigenvalues and eigenvectors describe the directions a matrix only stretches, and by how much. The GDC returns them from a matrix in one step. Use the results to find the long-term behaviour of a system, and check your answer by confirming that the matrix times each eigenvector equals the eigenvalue times that vector.
When you'd use this
Finding the long-run behaviour of a Markov chain or coupled system without repeatedly multiplying matrices.
A question directly asks for the eigenvalues and eigenvectors of a given matrix.
Diagonalising a matrix, e.g. to find a formula for its n-th power.
Checking eigenvalues found by hand from the characteristic equation.
At a glance: TI-84 Plus CE, Casio fx-CG50 and TI-Nspire CX compared
TI-84 Plus CE
Casio fx-CG50 and fx-CG100
TI-Nspire CX
Key sequence
2nd → MATRX → EDIT to enter [A]. On the home screen: eigVl([A]) gives eigenvalues; eigVc([A]) gives eigenvectors as columns. These are in the MATRX → MATH menu.
Run-Matrix → MAT to enter the matrix; then OPTN → MAT → EIG → EigenVal / EigenVec.
Enter the matrix, then menu → Matrix & Vector → Eigenvalues (or Eigenvectors). Or type eigVl(A) and eigVc(A) directly.
On a TI-84 Plus CE
Enter the square matrix (2×2 or 3×3) into the calculator.
2nd → MATRX → EDIT to enter [A]. On the home screen: eigVl([A]) gives eigenvalues; eigVc([A]) gives eigenvectors as columns. These are in the MATRX → MATH menu.
Each eigenvalue λ has a corresponding eigenvector column. Check: A × eigenvector = λ × eigenvector.
For a 2×2 transition matrix, the dominant eigenvalue is 1 and its eigenvector gives the long-run steady state (normalise so the components sum to 1).
Tip: Eigenvectors from the GDC may be scaled differently - only the direction matters, not the magnitude. Always normalise if you need probabilities or proportions.
On a Casio fx-CG50 and fx-CG100
Enter the square matrix (2×2 or 3×3) into the calculator.
Run-Matrix → MAT to enter the matrix; then OPTN → MAT → EIG → EigenVal / EigenVec.
Each eigenvalue λ has a corresponding eigenvector column. Check: A × eigenvector = λ × eigenvector.
For a 2×2 transition matrix, the dominant eigenvalue is 1 and its eigenvector gives the long-run steady state (normalise so the components sum to 1).
Tip: Eigenvectors from the GDC may be scaled differently - only the direction matters, not the magnitude. Always normalise if you need probabilities or proportions.
On a TI-Nspire CX
Enter the square matrix (2×2 or 3×3) into the calculator.
Enter the matrix, then menu → Matrix & Vector → Eigenvalues (or Eigenvectors). Or type eigVl(A) and eigVc(A) directly.
Each eigenvalue λ has a corresponding eigenvector column. Check: A × eigenvector = λ × eigenvector.
For a 2×2 transition matrix, the dominant eigenvalue is 1 and its eigenvector gives the long-run steady state (normalise so the components sum to 1).
Tip: Eigenvectors from the GDC may be scaled differently - only the direction matters, not the magnitude. Always normalise if you need probabilities or proportions.
When would I need to find eigenvalues and eigenvectors in IB Maths?
Used in AI HL for long-run behaviour of systems (e.g. coupled populations, Markov chains). The GDC computes them directly - no characteristic polynomial by hand. Finding the long-run behaviour of a Markov chain or coupled system without repeatedly multiplying matrices. A question directly asks for the eigenvalues and eigenvectors of a given matrix.
How do I find eigenvalues and eigenvectors on a TI-84 Plus CE?
1. Enter the square matrix (2×2 or 3×3) into the calculator. 2. 2nd → MATRX → EDIT to enter [A]. On the home screen: eigVl([A]) gives eigenvalues; eigVc([A]) gives eigenvectors as columns. These are in the MATRX → MATH menu. 3. Each eigenvalue λ has a corresponding eigenvector column. Check: A × eigenvector = λ × eigenvector. 4. For a 2×2 transition matrix, the dominant eigenvalue is 1 and its eigenvector gives the long-run steady state (normalise so the components sum to 1).
How do I find eigenvalues and eigenvectors on a Casio fx-CG50 and fx-CG100?
1. Enter the square matrix (2×2 or 3×3) into the calculator. 2. Run-Matrix → MAT to enter the matrix; then OPTN → MAT → EIG → EigenVal / EigenVec. 3. Each eigenvalue λ has a corresponding eigenvector column. Check: A × eigenvector = λ × eigenvector. 4. For a 2×2 transition matrix, the dominant eigenvalue is 1 and its eigenvector gives the long-run steady state (normalise so the components sum to 1).
How do I find eigenvalues and eigenvectors on a TI-Nspire CX?
1. Enter the square matrix (2×2 or 3×3) into the calculator. 2. Enter the matrix, then menu → Matrix & Vector → Eigenvalues (or Eigenvectors). Or type eigVl(A) and eigVc(A) directly. 3. Each eigenvalue λ has a corresponding eigenvector column. Check: A × eigenvector = λ × eigenvector. 4. For a 2×2 transition matrix, the dominant eigenvalue is 1 and its eigenvector gives the long-run steady state (normalise so the components sum to 1).
What should I watch out for when I find eigenvalues and eigenvectors?
Eigenvectors from the GDC may be scaled differently - only the direction matters, not the magnitude. Always normalise if you need probabilities or proportions.
How are marks awarded when I find eigenvalues and eigenvectors in an IB exam?
In the worked example on this page (4 marks, Paper 2), the marks are: M1: Find the eigenvalues, λ = 5 and λ = 2; A1: Eigenvector (1, 1) for λ = 5; A1: Eigenvector (1, −2) for λ = 2.
