Theory of Statistics - George Mason University

0.3.2.8 Spectral Decomposition
0.3 Some Basics of Linear Algebra 787
For a symmetric matrix A, we can always write A = VCV T , as above. This is
called the spectral decomposition, and is unique except for the ordering and
the choice of eigenvectors for eigenvalues with multiplicities greater than 1.
We can also write
A =
ciPi,
where the Pi are the outer products of the eigenvectors,
and are called spectral projectors.
0.3.2.9 Matrix Norms
i
Pi = viv T i ,
A matrix norm is generally required to satisfy one more property in addition to
those listed above for the definition of a norm. It is the consistency property:
AB ≤ A B. The Lp matrix norm for the n × m matrix A is defined as
Ap = max
xp=1 Axp.
The L2 matrix norm has the interesting relationship
A2 = ρ(ATA), where ρ(·) is the spectral radius (the modulus of the eigenvalue with the
maximum modulus).
The “usual” matrix norm is the Frobenius norm:
AF =
i,j
a 2 ij .
0.3.2.10 Idempotent and Projection Matrices
A matrix A such that AA = A is called an idempotent matrix. An idempotent
matrix is square, and it is either singular or it is the identity matrix. (It must
be square in order to be conformable for the indicated multiplication. If it
is not singular, we have A = (A −1 A)A = A −1 (AA) = A −1 A = I; hence, an
idempotent matrix is either singular or it is the identity matrix.)
If A is idempotent and n × n, then (I − A) is also idempotent, as we see
by multiplication.
In this case, we also have
Theory of Statistics c○2000–2013 James E. Gentle