Matrices: a brief overview - Signal Processing and Optimization for Transceiver Systems - page 758

4. P has an eigenvalue equal to zero if and only if it is singular (determinant

equal to zero).

T

5. P and P have the same set of eigenvalues including multiplicity.

6. For a (lower or upper) triangular matrix, the eigenvalues are equal to the
diagonal elements. Diagonal matrices also have this property.

7. Even if P = 0 , it is possible for all eigenvalues to be zero. For example,

⎡ ⎤

012
004
000

⎣ ⎦

P = (B . 11)

has all eigenvalues equal to zero.

8. The determinant and trace of an M × M matrix P are related to its M

eigenvalues λ k as follows:

M− 1 M− 1

det P = λ k and Tr ( P )= λ k . (B . 12)

k =0 k =0

9. For nonsingular P , the eigenvalues of P − 1 are reciprocals of those of P .

10. If λ k are the eigenvalues of P , the eigenvalues of P + σ I are λ k + σ .

11. The matrix T − 1 PT has the same set of eigenvalues as P (including mul-
tiplicity). This is true for any nonsingular T . The matrix T − 1 PT is said
to be a similarity transformation of P .

B.4.1 Invertible square matrices

From the preceding discussions we see that for an M × M matrix P the following

statements are equivalent:

P − 1 exists. 1.

2. P is nonsingular.

3. [det P ] =0 .

4. All eigenvalues of P are nonzero.

5. There is no nonzero vector v that annihilates P (i.e., makes Pv = 0 ).

6. The rank of P is M .

7. The M columns of P are linearly independent (and so are the rows).