§1.2 Elementary Algebra

where $\mathrm{\ell }$ is $n$ or $n-1$ according as $n$ is even or odd.

In (1.2.6)–(1.2.9) $k$ and $m$ are nonnegative integers and $z$ is complex.

See also §26.3.

where $\mathrm{\ell }$ = last term of the series = $a+(n-1)d$.

The full index form of an $m\times n$ matrix $𝐀$ is

with matrix elements $a_{ij}\in ℂ$, where $i$, $j$ are the row and column indices, respectively. A matrix is zero if all its elements are zero, denoted $𝟎$. A matrix is real if all its elements are real.

The transpose of $𝐀$ = $[a_{ij}]$ is the $n\times m$ matrix

the complex conjugate is

the Hermitian conjugate is

Multiplication by a scalar is given by

For matrices $𝐀$, $𝐁$ and $𝐂$ of the same dimensions,

Multiplication of an $m\times n$ matrix $𝐀$ and an $m^{\prime }\times n^{\prime }$ matrix $𝐁$, giving the $m\times n^{\prime }$ matrix $𝐂=𝐀𝐁$ is defined iff $n=m^{\prime }$. If defined, $𝐂=[c_{ij}]$ with

This is the row times column rule.

Assuming the indicated multiplications are defined: matrix multiplication is associative

distributive if $𝐁$ and $𝐂$ have the same dimensions

The transpose of the product is

All of the above are defined for $n\times n$, 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 $n$ is an $n\times 1$ matrix

and the corresponding transposed row vector of length $n$ is

The column vector $𝐯$ is often written as ${[v_1{v}_2\mathrm{\dots }{v}_n]}^{\mathrm{T}}$ to avoid inconvenient typography. The zero vector $𝐯=𝟎$ has $v_i=0$ for $i=1,2,\mathrm{\dots },n$.

Column vectors $𝐮$ and $𝐯$ of the same length $n$ have a scalar product

The dot product notation $𝐮\cdot 𝐯$ is reserved for the physical three-dimensional vectors of (1.6.2).

The scalar product has properties

for $\alpha ,\beta \in ℂ$

and

if and only if $𝐯=𝟎$.

If $𝐮$, $𝐯$, $\alpha$ and $\beta$ are real the complex conjugate bars can be omitted in (1.2.40)–(1.2.42).

Two vectors $𝐮$ and $𝐯$ are orthogonal if

The $l^p$ norm of a (real or complex) vector is

Special cases are the Euclidean length or $l^2$ norm

the $l^1$ norm

and as $p\to \mathrm{\infty }$

The $l^2$ norm is implied unless otherwise indicated. A vector of $l^2$ norm unity is normalized and every non-zero vector $𝐯$ can be normalized via $𝐯\to 𝐯/\Vert 𝐯\Vert$.

If

we have Hölder’s Inequality

which for $p=q=2$ is the Cauchy-Schwartz inequality

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

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

The identity matrix $𝐈$, is defined as

A matrix $𝐀$ is: a diagonal matrix if

a real symmetric matrix if

an Hermitian matrix if

a tridiagonal matrix if

$𝐀$ is an upper or lower triangular matrix if all $a_{ij}$ vanish for $i>j$ 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 $𝐀$ has a determinant, $det(𝐀)$, explored further in §1.3, denoted, in full index form, as

where $det(𝐀)$ is defined by the Leibniz formula

${𝔖}_n$ is the set of all permutations of the set $\{1,2,\mathrm{\dots },n\}$. See §26.13 for the terminology used herein.

If det($𝐀$) $\ne$ $0$, $𝐀$ has a unique inverse, ${𝐀}^{-1}$, such that

A square matrix $𝑨$ is singular if $det(𝐀)=0$, otherwise it is non-singular. If $det(𝐀)=0$ then $𝐀𝐁=𝐀𝐂$ does not imply that $𝐁=𝐂$; if $det(𝐀)\ne 0$, then $𝐁=𝐂$, as both sides may be multiplied by ${𝐀}^{-1}$.

Given a square matrix $𝐀$ and a vector $𝐜$. If $det(𝐀)\ne 0$ the system of $n$ linear equations in $n$ unknowns,

has a unique solution, $𝐛={𝐀}^{-1}𝐜$. If $det(𝐀)=0$ then, depending on $𝐜$, 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 $𝐀𝐛=𝟎$. Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii).

The trace of $𝐀=[a_{ij}]$ is

Further,

and

If $𝐀𝐁=𝐁𝐀$ the matrices $𝐀$ and $𝐁$ are said to commute. The difference between $𝐀𝐁$ and $𝐁𝐀$ is the commutator denoted as

Let $\Vert 𝐱\Vert ={\Vert 𝐱\Vert }_2$ the $l^2$ norm, and ${𝐄}_n$ the space of all $n$-dimensional vectors. We take ${𝐄}_n\subset {ℂ}^n$, but we can also restrict ourselves to vectors and matrices with only real elements. The norm of an order $n$ square matrix, $𝐀$, is

Then

and

A square matrix $𝐀$ has an eigenvalue $\lambda$ with corresponding eigenvector $𝐚\ne 𝟎$ if

Here $𝐚$ and $\lambda$ may be complex even if $𝐀$ 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 ${𝐯}_1,\mathrm{\dots },{𝐯}_n$ are linearly independent if ${\sum }_{i=1}^n{c}_i{𝐯}_i=𝟎$ implies that all coefficients $c_i$ are zero. A matrix $𝐀$ of order $n$ is non-defective if it has $n$ linearly independent (possibly complex) eigenvectors, otherwise $𝐀$ is called defective. Non-defective matrices are precisely the matrices which can be diagonalized via a similarity transformation of the form