- Use of terms singular, diagonal, unit, null, symmetric.
- Operations of addition, subtraction, multiplication, inverse and
[We will use for the transpose of .]
- The trace of a matrix , written tr, is defined as the sum
of the diagonal elements of A. That is,
- Linear Independence and Rank
- Let , ..., be a set of vectors and , ...,
be scalar constants. If
, the the set of vectors is linearly independent.
- The rank of a set of vectors is the maximum number of linearly
independent vectors in the set.
- For a square matrix , the rank of , denoted by r, is the
maximum order of non-zero subdeterminants.
- Quadratic Forms
For a p-vector x, where
, and a square
is a quadratic form in
The matrix and the quadratic form are called:
- positive semidefinite if
- positive definite if
- A necessary and sufficient condition for to be positive definite
is that each leading diagonal sub-determinant is greater than . So a
positive definite matrix is non-singular.
- A necessary and sufficient condition for a symmetric
matrix to be positive definite is that there exists a non-singular
matrix such that .
A matrix is said to be orthogonal if (or ).
- An orthogonal matrix is non-singular.
- The determinant of an orthogonal matrix is .
- The transpose of an orthogonal matrix is also orthogonal.
- The product of two orthogonal matrices is orthogonal.
- If is orthogonal, trtrtr.
- If is orthogonal, rr.
- Eigenvalues and eigenvectors.
Eigenvalues of a square matrix are defined as the roots of the equation
. The corresponding x satisfying
are the eigenvectors.
- The eigenvectors corresponding to two different eigenvalues are
- The number of non-zero eigenvalues of a square matrix
is equal to the rank of .
- Reduction to diagonal form
- Given any symmetric matrix there exists an orthogonal matrix P
such that where is a diagonal matrix whose elements
are the eigenvalues of . We write
- If is not of full rank, some of the will be zero.
- If is positive definite (and therefore non-singular), all
the will be greater than zero.
- The eigenvectors of form the columns of matrix .
- If is symmetric of rank and is orthogonal such that
- For every quadratic form
there exists an
which reduces Q to a diagonal
quadratic form so that
where is the rank of .
- Idempotent Matrices.
A matrix is said to be idempotent if . In the following
we shall mean symmetric idempotent matrices. Some properties are:
- If A is idempotent and non-singular then . To prove this,
note that and pre-multiply both sides by .
- The eigenvalues of an idempotent matrix are either or .
- If is idempotent of rank , there exists an orthogonal
matrix such that where is a diagonal matrix with
the first leading diagonal elements and the remainder .
- If is idempotent of rank then tr. To prove this,
note that there is an orthogonal matrix such that . Now
- If the ith diagonal element of is zero, all elements in the
ith row and column are zero.
- All idempotent matrices not of full rank are positive
semi-definite. No idempotent matrix can have negative elements on its