Consider the special form of the linear system (C.13) in
which the right-hand side vector y is a multiple

of the solution vector x:

Ax = λx, (C.48)

or, written in full,

This is called the standard (or classical) algebraic
eigenproblem. System (C.48) can be rearranged into

the homogeneous form

A nontrivial solution of this equation is possible if and
only if the coefficient matrix A−λI is singular.

Such a condition can be expressed as the vanishing of the determinant

When this determinant is expanded, we obtain an algebraic polynomial equation in λ of degree n:

This is known as the characteristic equation of the matrix
A. The left-hand side is called the characteristic

polynomial. We known that a polynomial of degree n has n (generally complex)
roots λ_{1}, λ_{2},

. . ., λ_{n}. These n numbers are called the eigenvalues, eigenroots or
characteristic values of matrix A.

With each eigenvalue λ_{i} there is an associated vector x_{i}
that satisfies

This x_{i} is called an eigenvector or
characteristic vector. An eigenvector is unique only up to a scale

factor since if x_{i} is an eigenvector, so is βx_{i} where β is
an arbitrary nonzero number. Eigenvectors are

often normalized so that their Euclidean length is 1, or their largest component
is unity.

Real symmetric matrices are of special importance in the
finite element method. In linear algebra

books dealing with the algebraic eigenproblem it is shown that:

(a) The n eigenvalues of a real symmetric matrix of order n are real.

(b) The eigenvectors corresponding to distinct eigenvalues are orthogonal. The
eigenvectors corresponding

to multiple roots may be orthogonalized with respect to each other.

(c) The n eigenvectors form a complete orthonormal basis for the Euclidean space
E_{n}.

Let A be an n × n square symmetric matrix. A is said to be positive definite (p.d.) if

A positive definite matrix has rank n. This property can
be checked by computing the n eigenvalues

λ_{i} of Az = λz. If all λ_{i} > 0, A is p.d.

A is said to be nonnegative if zero equality is allowed in (C.54):

A p.d. matrix is also nonnegative but the converse is not
necessarily true. This property can be checked

by computing the n eigenvalues λ_{i} of Az = λz. If r eigenvalues λ_{i}
> 0 and n−r eigenvalues are zero,

A is nonnegative with rank r .

A symmetric square matrix A that has at least one negative eigenvalue is called indefinite.

Let A be an n × n real square matrix. This matrix is called normal if

A normal matrix is called orthogonal if

All eigenvalues of an orthogonal matrix have modulus one,
and the matrix has rank n.

The generalization of the orthogonality property to complex matrices, for which
transposition is replaced

by conjugation, leads to unitary matrices. These are not required, however, for
the material

covered in the text.

The Sherman-Morrison formula gives the inverse of a matrix
modified by a rank-one matrix. The Woodbury

formula extends the Sherman-Morrison formula to a modification of arbitrary
rank. In structural analysis these

formulas are of interest for problems of structural modifications, in which a
finite-element (or, in general, a discrete

model) is changed by an amount expressable as a low-rank correction to the
original model.

Let A be a square n × n invertible matrix, whereas u and v
are two n-vectors and β an arbitrary scalar. Assume

that
Then

This is called the Sherman-Morrison formula when β = 1.
Since any rank-one correction to A can be written as

, (C.58) gives the rank-one change to its inverse. The proof is by direct
multiplication as in Exercise C.5.

For practical computation of the change one solves the linear systems Aa = u and
Ab = v for a and b, using the

known
. Compute
. If
, the change to A^{−1} is the dyadic

Let again A be a square n × n invertible matrix, whereas U
and V are two n × k matrices with k ≤ n and β an

arbitrary scalar. Assume that the k ×k matrix
in which
denotes the k ×k identity matrix,

is invertible. Then

This is called the Woodbury formula. It reduces to (C.58)
if k = 1, in which case Σ ≡ σ is a scalar. The proof

is by direct multiplication.

Let denote the adjoint of A. Taking the determinants from both sides of one obtains

If A is invertible, replacing this becomes

Similarly, one can show that if A is invertible, and U and V are n × k matrices,

**Exercises for Appendix C: Determinants, Inverses,
Eigenvalues**

**EXERCISE C.1**

If A is a square matrix of order n and c a scalar, show that det(cA) = c^{n}
detA.

**EXERCISE C.2**

Let u and v denote real n-vectors normalized to unit
length, so that
= u = 1 and
= 1, and let I denote the

n × n identity matrix. Show that

**EXERCISE C.3**

Let u denote a real n-vector normalized to unit length, so that
= u = 1 and I denote the n ×n identity matrix.

Show that

is orthogonal: H^{T}H = I, and idempotent: H^{2}
= H. This matrix is called a elementary Hermitian, a Householder

matrix, or a reflector. It is a fundamental ingredient of many linear algebra
algorithms; for example the QR

algorithm for finding eigenvalues.

**EXERCISE C.4**

The trace of a n × n square matrix A, denoted trace(A) is the sum
a_{ii} of its diagonal coefficients. Show

that if the entries of A are real,

**EXERCISE C.5**

Prove the Sherman-Morrison formula (C.59) by direct matrix multiplication

**EXERCISE C.6**

Prove the Sherman-Morrison formula (C.59) for β = 1 by considering the following
block bordered system

in which I_{k} and I_{n} denote the identy
matrices of orders k and n, respectively. Solve (C.62) two ways: eliminating

first B and then C, and eliminating first C and then B. Equate the results for
B.

**EXERCISE C.7**

Show that the eigenvalues of a real symmetric square matrix are real, and that
the eigenvectors are real vectors.

**EXERCISE C.8**

Let the n real eigenvalues λ_{i} of a real n × n symmetric matrix A be
classified into two subsets: r eigenvalues are

nonzero whereas n − r are zero. Show that A has rank r .

**EXERCISE C.9**

Show that if A is p.d., Ax = 0 implies that x = 0.

**EXERCISE C.10**

Show that for any real m × n matrix A, A^{T}A exists and is nonnegative.

**EXERCISE C.11**

Show that a triangular matrix is normal if and only if it is diagonal.

**EXERCISE C.12**

Let A be a real orthogonal matrix. Show that all of its eigenvalues λ_{i}
, which are generally complex, have unit

modulus.

**EXERCISE C.13**

Let A and T be real n × n matrices, with T nonsingular. Show that T^{−1}AT
and A have the same eigenvalues.

(This is called a similarity transformation in linear algebra).

**EXERCISE C.14**

(Tough) Let A be m × n and B be n × m. Show that the nonzero eigenvalues of AB
are the same as those of BA

(Kahan).

**EXERCISE C.15**

Let A be real skew-symmetric, that is, A = −A^{T} . Show that all
eigenvalues of A are purely imaginary or zero.

**EXERCISE C.16**

LetAbe real skew-symmetric, that is,
Showthatcalled a Cayley transformation,

is orthogonal.

**EXERCISE C.17**

Let P be a real square matrix that satisfies

Such matrices are called idempotent, and also orthogonal
projectors. Show that all eigenvalues of P are either

zero or one.

**EXERCISE C.18**

The necessary and sufficient condition for two square matrices to commute is
that they have the same eigenvectors.

**EXERCISE C.19**

A matrix whose elements are equal on any line parallel to the main diagonal is
called a Toeplitz matrix. (They

arise in finite difference or finite element discretizations of regular
one-dimensional grids.) Show that if T_{1} and

T_{2} are any two Toeplitz matrices, they commute: T_{1}T_{2}
= T_{2}T_{1}. Hint: do a Fourier transform to show that the

eigenvectors of any Toeplitz matrix are of the form
then apply the previous Exercise.