Identity and diagonal matrices
Square matrices are matrices that have the same number of rows as columns. The following are important instances of square matrices.
The identity matrix (often denoted , or simply , if context allows), has ones on its diagonal and zeros elsewhere. It is square, diagonal and symmetric. This matrix satisfies for every matrix with columns, and for every matrix with rows.Matlab syntax
>> I3 = eye(3); % the 3x3 identity matrix >> A = eye(3,4); % a 3x4 matrix having the 3x3 identity in its first 3 columns
Diagonal matrices are square matrices with when . A diagonal matrix can be denoted as , with the vector containing the elements on the diagonal. We can also write
where by convention the zeros outside the diagonal are not written.Matlab syntax
>> A = diag([1 2 3]); % a diagonal matrix with 1,2,3 on the diagonal >> A = spdiags([1 2 3]',0,3,3); % the same matrix declared as a sparse matrix
Symmetric matrices are square matrices that satisfy for every pair . An entire section is devoted to symmetric matrices.
A square matrix is upper triangular if when . Here are a few examples:
A matrix is lower triangular if its transpose is upper triangular. For example:
Orthogonal (or, unitary) matrices are square matrices, such that the columns form an orthonormal basis. If is an orthogonal matrix, then
Thus, . Similarly, .
Orthogonal matrices correspond to rotations or reflections across a direction: they preserve length and angles. Indeed, for every vector ,
Thus, the underlying linear map preserves the length (measured in Euclidean norm). This is sometimes referred to as the rotational invariance of the Euclidean norm.
In addition, angles are preserved: if are two vectors with unit norm, then the angle between them satisfies , while the angle between the rotated vectors , satisfies . Since
we obtain that the angles are the same. (The converse is true: any square matrix that preserves lengths and angles is orthogonal.)
Geometrically, orthogonal matrices correspond to rotations (around a point) or reflections (around a line passing through the origin).
Dyads are a special class of matrices, also called rank-one matrices, for reasons seen later.
A matrix is a dyad if it is of the form for some vectors , . The dyad acts on an input vector as follows:
In terms of the associated linear map, for a dyad, the output always points in the same direction in output space (), no matter what the input is. The output is thus always a simple scaled version of . The amount of scaling depends on the vector , via the linear function .
We can always normalize the dyad, by assuming that both are of unit (Eculidean) norm, and using a factor to capture their scale. That is, any dyad can be written in normalized form:
where , and .