Find the eigenvalues and eigenvectors of a 2 by 2 or 3 by 3 matrix: build the characteristic polynomial, solve it, find the directions the matrix does not turn, and check every one by multiplying.
Find the eigenvalues and eigenvectors of this matrix.
A=1000−2020−1
The result
λ1=−2:010,λ2=−1:10−1,λ3=1:100
Three different eigenvalues, each with its own direction.
checked49 lines of this solution were evaluated with exact arithmetic before it was shown to you. A solution with a false line is never served.
Picture 1 of 2Step 1: Say what we are looking for
The matrix whose own directions we are hunting
1020−2000−1
A matrix turns most vectors off their own line. The question this page answers is which lines it does NOT turn: on those, the matrix only stretches or shrinks, and the amount it stretches by is the eigenvalue.
1
Say what we are looking for
Why this step
A matrix moves almost every vector to a new direction. The exceptions are rare and precious: directions the matrix does NOT turn, only stretches. Along such a direction the whole complicated matrix behaves like multiplication by a single number. Find those directions and the matrix stops being a table of numbers and becomes something you can see.
A=1000−2020−1,A−λI=1−λ000−2−λ020−1−λ
Subtracting λI takes λ off each diagonal entry and leaves the rest alone.
The matrix whose own directions we are hunting
1020−2000−1
A matrix turns most vectors off their own line. The question this page answers is which lines it does NOT turn: on those, the matrix only stretches or shrinks, and the amount it stretches by is the eigenvalue.
2
Build the characteristic polynomial
Why this step
We expand det(A−λI) with λ left as a letter. The result is a polynomial in λ of degree 3, called the characteristic polynomial, and its roots are the eigenvalues. Two of its coefficients can be predicted before any work: the sum of the diagonal of A (the trace) and detA. They make a free check on the algebra.
det(A−λI)=(1−λ)((−2−λ)(−1−λ)−0⋅0)+2(0⋅0−(−2−λ)⋅0)
Expand along the first row: each first-row entry times the 2×2 determinant left when its row and column are struck out, with signs +,−,+. Entries that are 0 contribute nothing and are left out. checked
=(1−λ)(λ2+3λ+2)+2(0)
Work out each 2×2 determinant and collect its powers of λ. checked
=(−λ3−2λ2+λ+2)+(0)
Multiply each first-row entry into its bracket. checked
=−λ3−2λ2+λ+2
Remove the brackets, remembering that a minus sign in front reaches every term inside, and collect like powers of λ. checked
The matrix, before lambda is subtracted from its diagonal
1020−2000−1
Subtracting lambda down the diagonal is what turns the question into one about a determinant: we want the values of lambda that flatten this matrix, because only a flattened matrix can send a non-zero vector to zero.