Math 778S Spectral Graph Theory
Handout #1: Basic linear algebra
Let Abe an n×n-matrix. If there exists a non-zero column vector αand a
number λsatisfying
Aα =λα,
we say λis an eigenvalue of Aand αis an eigenvector of Acorresponding to the
eigenvalue λ. For any c6= 0, cα is also an eigenvector of A. We can normalize
it so that αbecomes a unit vector. Note that αand λis NOT necessary to be
real even if Ais a real matrix.
All eigenvalues λ1,λ2, . . . , λnare the root of the characteristic polynomial
det(λI −A) =
n
Y
i=1
(λ−λi).
By comparing the coefficient, we have
n
Y
i=1
λi= det(A)
n
X
i=1
λi= tr(A).
Here tr(A) = Pi=1 aii is called the trace of a matrix.
Let f(x) be any polynomial of x. It is easy to check that
f(A)α=f(λ)α.
In conclusion, f(A) has eigenvalues f(λ1), f(λ2), . . . , f (λn). In particular, we
have n
X
i=1
λk
i= tr(Ak).
From now on, we assume Ais a symmetric real matrix (i.e., A0=A). Then
all eigenvalues are real and all eigenvectors can be made into an orthogonal
base. Two eigenvectors corresponding to the distinct eigenvalues are always or-
thogonal to each other. Two eigenvectors corresponding to the same eigenvalue
is not always orthogonal to each other.
A matrix Ois orthogonal if O0=O−1. There exists an orthogonal matrix
Osatisfying
AO =OΛ.
Here Λ is a diagonal matrix with diagonal entries λ1,λ2, . . . , λn. Let αibe the
i-th column of O. Then αiis the eigenvector corresponding to λi. I.e.,
Aαi=λiαi.
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