§1.2 Elementary Algebra

where is or according as is even or odd.

In (1.2.6)–(1.2.9) and are nonnegative integers and is complex.

See also §26.3.

where = last term of the series = .

The full index form of an matrix is

with matrix elements , where , are the row and column indices, respectively. A matrix is zero if all its elements are zero, denoted . A matrix is real if all its elements are real.

The transpose of = is the matrix

the complex conjugate is

the Hermitian conjugate is

Multiplication by a scalar is given by

For matrices , and of the same dimensions,

Multiplication of an matrix and an matrix , giving the matrix is defined iff . If defined, with

This is the row times column rule.

Assuming the indicated multiplications are defined: matrix multiplication is associative

distributive if and have the same dimensions

The transpose of the product is

All of the above are defined for , or square matrices of order n, note that matrix multiplication is not necessarily commutative; see §1.2(vi) for special properties of square matrices.

A column vector of length is an matrix

and the corresponding transposed row vector of length is

The column vector is often written as to avoid inconvenient typography. The zero vector has for .

Column vectors and of the same length have a scalar product

The dot product notation is reserved for the physical three-dimensional vectors of (1.6.2).

The scalar product has properties

for

and

if and only if .

If , , and are real the complex conjugate bars can be omitted in (1.2.40)–(1.2.42).

Two vectors and are orthogonal if

The norm of a (real or complex) vector is

Special cases are the Euclidean length or norm

the norm

and as

The norm is implied unless otherwise indicated. A vector of norm unity is normalized and every non-zero vector can be normalized via .

If

we have Hölder’s Inequality

which for is the Cauchy-Schwartz inequality

the equality holding iff is a scalar (real or complex) multiple of . The triangle inequality,

For similar and more inequalities see §1.7(i).

The identity matrix , is defined as

A matrix is: a diagonal matrix if

a real symmetric matrix if

an Hermitian matrix if

a tridiagonal matrix if

is an upper or lower triangular matrix if all vanish for or , respectively.

Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix.

The matrix has a determinant, , explored further in §1.3, denoted, in full index form, as

where is defined by the Leibniz formula

is the set of all permutations of the set . See §26.13 for the terminology used herein.

If det() , has a unique inverse, , such that

A square matrix is singular if , otherwise it is non-singular. If then does not imply that ; if , then , as both sides may be multiplied by .

Given a square matrix and a vector . If the system of linear equations in unknowns,

has a unique solution, . If then, depending on , there is either no solution or there are infinitely many solutions, being the sum of a particular solution of (1.2.61) and any solution of . Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii).

The trace of is

Further,

and

If the matrices and are said to commute. The difference between and is the commutator denoted as

Let the norm, and the space of all -dimensional vectors. We take , but we can also restrict ourselves to vectors and matrices with only real elements. The norm of an order square matrix, , is

Then

and

A square matrix has an eigenvalue with corresponding eigenvector if

Here and may be complex even if is real. Eigenvalues are the roots of the polynomial equation

and for the corresponding eigenvectors one has to solve the linear system

Numerical methods and issues for solution of (1.2.72) appear in §§3.2(iv) to 3.2(vii).

Nonzero vectors are linearly independent if implies that all coefficients are zero. A matrix of order is non-defective if it has linearly independent (possibly complex) eigenvectors, otherwise is called defective. Non-defective matrices are precisely the matrices which can be diagonalized via a similarity transformation of the form

The columns of the invertible matrix are eigenvectors of , and is a diagonal matrix with the eigenvalues as diagonal elements. The diagonal elements are not necessarily distinct, and the number of identical (degenerate) diagonal elements is the multiplicity of that specific eigenvalue. The sum of all multiplicities is .

For non-defective we obtain from (1.2.73) and (1.3.7)

Thus is the product of the (counted according to their multiplicities) eigenvalues of . Similarly, we obtain from (1.2.73) and (1.2.65)

Thus is the sum of the (counted according to their multiplicities) eigenvalues of .

The matrix exponential is defined via

which converges, entry-wise or in norm, for all .

It follows from (1.2.73), (1.2.74) and (1.2.75) that, for a non-defective matrix ,

Formula (1.2.77) is more generally valid for all square matrices , not necessarily non-defective, see Hall (2015, Thm 2.12).