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184 ALGEBRA

Properties of spectrum of a matrices:
1. Similar matrices have the same spectrum.
¯
2. If λ is an eigenvalue of a normal matrix A (see Paragraph 5.2.1-3), then λ is an eigenvalue
1
of the matrix A ;Re λ is an eigenvalue of the matrix H 1 = (A + A ); and Im λ is an
∗
∗
2
1
eigenvalue of the matrix H 2 = 2i (A – A ).
∗
3. All eigenvalues of a normal matrix are real if and only if this matrix is similar to a
Hermitian matrix.
4. All eigenvalues of a unitary matrix have absolute values equal to 1.
5. A square matrix is nondegenerate if and only if all its eigenvalues are different from
zero.
A nonzero (column) vector X (see Paragraphs 5.2.1-1 and 5.2.1-2) satisfying the con-
dition
AX = λX
is called an eigenvector of the matrix A corresponding to the eigenvalue λ. Eigenvectors
corresponding to distinct eigenvalues of A are linearly independent.

5.2.3-6. Reduction of a square matrix to triangular form.
–1
THEOREM. For any square matrix A there exists a similarity transformation A = S AS
2
such that A is a triangular matrix.
2
The diagonal entries of any triangular matrix similar to a square matrix A of size n × n
coincide with the eigenvalues of A; each eigenvalue λ i of A occurs m ≥ 1 times on the

i
diagonal. The positive integer m is called the algebraic multiplicity of the eigenvalue λ i .

i

Note that m = n.
i
i
The trace Tr(A) is equal to the sum of all eigenvalues of A, each eigenvalue counted
according to its multiplicity, i.e.,

Tr(A)= m λ i .
i
i
The determinant det A is equal to the product of all eigenvalues of A, each eigenvalue
counted according to its multiplicity, i.e.,
m
det A = λ i i .
i

5.2.3-7. Reduction of a square matrix to diagonal form.
THEOREM 1. If A is a square matrix similar to some normal matrix, then there is a
–1
similarity transformation A = S AS such that the matrix A is diagonal.
2
2
THEOREM 2. Two Hermitian matrices A and B can be reduced to diagonal form by the
same similarity transformation if and only if AB = BA.
THEOREM 3. For any Hermitian matrix A, there is a nondegenerate matrix S such that
A = S AS is a diagonal matrix. The entries of A are real.
∗
2
2
THEOREM 4. For any real symmetric matrix A, there is a real nondegenerate matrix T
T
such that A = S AS is a diagonal matrix.
2