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What the lecture covers
For a 2×2 matrix, the usual method finds eigenvalues by forming the characteristic polynomial and solving a quadratic. The lecture presents a more direct route based on two facts: the trace is the sum of the eigenvalues, and the determinant is their product. Thus, if their mean is m and their product is p, the eigenvalues are m ± √(m² − p). For a matrix, m is half the sum of its diagonal entries, while p is its determinant.
Examples show how to apply the formula directly: the matrix with diagonal entries 8 and 6 and determinant 40 has eigenvalues 4 and 10; another matrix gives 2 ± √5. The lecture also considers the Pauli spin matrices, whose eigenvalues are ±1, and a normalized linear combination of them, for which the same result is easier to obtain from the mean and product than by expanding a characteristic polynomial. The shortcut is not a different mathematical method: it uses the same information as the quadratic equation, but makes the roles of trace and determinant explicit and avoids writing out the polynomial.
Key ideas
An eigenvalue λ makes A − λI send a nonzero vector to zero, so the determinant of A − λI must vanish.
The standard method expands the characteristic polynomial and solves the resulting quadratic.
For a 2×2 matrix, the trace is the sum of the eigenvalues and the determinant is their product.
Two numbers with mean m and product p are m ± √(m² − p), giving a direct formula for the eigenvalues.
Applying the formula to a matrix requires only its diagonal-entry mean and determinant.
The Pauli spin matrices have eigenvalues +1 and −1, and the formula also simplifies calculations for their normalized linear combinations.
The shortcut solves the same quadratic problem as the characteristic-polynomial method but uses the trace and determinant directly.
Sample questions
For a 2×2 matrix A, its characteristic polynomial is det(A − λI), where I is the identity matrix. Why do the roots of this polynomial give the eigenvalues of A?
AEach root is an entry of A whose row sums to zero.
BThe roots make the determinant of A itself equal to zero.
CThe roots are the diagonal entries of A, regardless of its other entries.
DAt each root, A − λI is singular, so there is a nonzero vector v with Av = λv.
Show answer
Correct answer: D. A scalar λ is an eigenvalue exactly when A − λI has a nonzero vector in its null space, which happens when its determinant is zero.
For a 2×2 matrix, what information does the trace—the sum of its diagonal entries—provide about its two eigenvalues, without determining either eigenvalue individually?
AIt equals the sum of the two eigenvalues, and therefore also their mean after division by two.
BIt equals the product of the two eigenvalues, and therefore determines their mean.
CIt guarantees that each eigenvalue equals the mean of the two diagonal entries.
DIt equals the difference between the two eigenvalues, regardless of their sum.
Show answer
Correct answer: A. The trace encodes the eigenvalues’ sum, so dividing that sum by two gives their mean; it does not specify each value on its own.