Fundamentals of Matrix Algebra, 2011a

4.1 Eigenvalues and Eigenvectors
. Example 85 Find ⃗x such that A⃗x = 5⃗x, where
[ ] 1 4
A = .
2 3
S
Recall that our algebra from before showed that if
A⃗x = λ⃗x then (A − λI)⃗x = ⃗0.
Therefore, we need to solve the equaon (A − λI)⃗x = ⃗0 for ⃗x when λ = 5.
[ ] [ ]
1 4 1 0
A − 5I = − 5
2 3 0 1
[ ] −4 4
=
2 −2
To solve (A − 5I)⃗x = ⃗0, we form the augmented matrix and put it into reduced row
echelon form: [ ]
[ ]
−4 4 0 −→ 1 −1 0
rref
.
2 −2 0
0 0 0
Thus
x 1 = x 2
x 2 is free
and
⃗x =
[ ] [ ]
x1 1
= x
x 2 .
2 1
We have [ infinite ] soluons to the equaon A⃗x = 5⃗x; any nonzero scalar mulple of the
1
vector is a soluon. We can do a few examples to confirm this:
1
[ ] [ ] 1 4 2
=
2 3 2
[ ] [ ] 1 4 7
=
2 3 7
] [ ] −3
=
−3
[ 1 4
2 3
[ ] [ ]
10 2
= 5 ;
10 2
[ ] [ ]
35 7
= 5 ;
35 7
]
[ −15
−15
[ ] −3
.
−3
= 5
.
Our method of finding the eigenvalues of a matrix A boils down to determining
which values of λ give the matrix (A−λI) a determinant of 0. In compung det (A − λI),
we get a polynomial in λ whose roots are the eigenvalues of A. This polynomial is important
and so it gets its own name.
167