Trace and determinant check
The eigenvalues add up to the trace (sum of diagonal entries) and multiply to the determinant. Use this to check your work.
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Analogy
A quick bank-statement check that the pieces add up to the total.
Example
Trace = 4 + 3 = 7 = 5 + 2.Determinant = 12 - 2 = 10 = 5 x 2.Both checks pass.So eigenvalues 5 and 2 are right.
In practice
Catches arithmetic slips and gives 2 by 2 eigenvalues quickly.
Practice
Flashcards
- The eigenvalues sum to which quantity?The trace of the matrix.
- The eigenvalues multiply to which quantity?The determinant.
Interview questions
- Find the eigenvalues and eigenvectors of A = [[4,1],[2,3]].
- Characteristic equation L^2 - 7L + 10 = 0
- Eigenvalues 5 and 2
- For 5: (A - 5I)v = 0 gives v = (1,1)
- For 2: gives v = (1,-2)
- Check using trace 7 and determinant 10
Questions people ask
Can a matrix have a zero eigenvalue?
Yes. If the determinant is zero, at least one eigenvalue is zero, since the eigenvalues multiply to the determinant. Such a matrix collapses some direction to nothing, so it has no inverse. A matrix is invertible exactly when none of its eigenvalues are zero.
How do I get a 2 by 2 matrix's eigenvalues from trace and determinant?
Solve L squared minus trace times L plus determinant equals zero. For trace 7 and determinant 10, that is L squared minus 7L plus 10, which factorises into L minus 5 and L minus 2. So the eigenvalues are 5 and 2.