Matrices

Matrix

A rectangular array of elements which may be real, complex, or functions.

Real Matrix

A matrix where all elements are real numbers.

Row Matrix

A matrix having only one row.

Column Matrix

A matrix having only one column.

Horizontal Matrix

A matrix where the number of columns exceeds the number of rows.

Vertical Matrix

A matrix where the number of rows exceeds the number of columns.

Null Matrix

A matrix where all elements are zero.

Square Matrix

A matrix with the same number of rows and columns.

Trace of a Matrix

The sum of the principal diagonal elements of a square matrix.

Symmetric Matrix

A matrix that is equal to its transpose.

Skew Symmetric Matrix

A matrix where the transpose is the negative of the original.

Diagonal Matrix

A matrix in which all elements outside the principal diagonal are zero.

Identity Matrix

A diagonal matrix in which all diagonal elements are 1.

Scalar Matrix

A diagonal matrix where all diagonal elements are equal.

Adjoint of Matrix

The transpose of the cofactor matrix.

Determinant

A scalar value that is a function of the entries of a square matrix.

Singular Matrix

A square matrix which does not have an inverse, i.e., its determinant is zero.

Non-Singular Matrix

A square matrix that has an inverse, i.e., its determinant is not zero.

Additive Inverse

Two matrices that add up to the zero matrix.

Matrix Multiplication

An operation where two matrices are multiplied to produce a third matrix, applicable when the number of columns in the first matrix equals the number of rows in the second matrix.

Commutative Property

A property stating that the order of addition or multiplication does not change the result.

Associative Property

A property that states that the way in which numbers are grouped does not change their sum or product.

Idempotent Matrix

A matrix that, when multiplied by itself, yields the same matrix.

Periodic Matrix

A matrix that returns to its original form after a finite number of multiplications by itself.

Nilpotent Matrix

A matrix for which a certain power of it (greater than zero) results in a null matrix.

Transpose of a Matrix

A new matrix whose rows are the columns of the original.

Symmetric Matrix Definition

A square matrix where A = A^T.

Skew Symmetric Matrix Definition

A square matrix where A = -A^T.

Cayley Hamilton Theorem

Every square matrix satisfies its own characteristic equation.