Types of Matrices

The matrix is positive definite if and only if the bilinear form , = is positive definite (and similarly for a positive definite sesquilinear form in the complex case). This is a coordinate realization of an inner product on a vector space.

Determinant of a Matrix. The determinant of a matrix is a special number that can be calculated from a square matrix.. A Matrix is an array of numbers:. A Matrix (This one has 2 Rows and 2 Columns)

Upper and Lower Triangular Matrices: { De nition: An upper triangular matrix is a square matrix in which all entries below the main diagonal are zero (only nonzero entries are found above the main diagonal - in the upper triangle).

The identity matrix also has the property that, when it is the product of two square matrices, the matrices can be said to be the inverse of one another. The identity matrix of a given size is the only idempotent matrix of that size having full rank.

When we multiply a matrix by its inverse we get the Identity Matrix (which is like "1" for matrices):

The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.

In other words, we can say that transpose of Matrix B is not equal to matrix B (). What is a Skew-Symmetric Matrix? Square Matrix A is said to be skew-symmetric if for all i and j. In other words, we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of Matrix A i.e ().

The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis. This characterization can be used to define the trace of a linear operator in general. Note that the trace is only defined for a square matrix (i.e., n × n).