Matrix

linear-algebra

Definition

Matrix

A matrix with coefficients , where is a field, is a rectangular array of coefficients:

where with and .
The set of all matrices with coefficients in is denoted by .

If , the the matrix is called square matrix.

Properties

Arithmetic

Addition

Let be two matrices:

Subtraction

Let be two matrices:

Multiplication

Let and be two matrices:

Example:

Transpose

Definition

Matrix Transpose

Let be a matrix over a field .

The transpose of , written , is the matrix obtained by swapping row and column elements.

Einstein Notation: .

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Power

Raising a matrix to the power of means matrix-multiplying matrix times:

Rank

Definition

Rank (Matrix)

The rank of a matrix is the dimension (number of basis vectors) of the linear span of its column or row vectors.

To determine the rank of a matrix, the Gaussian elimination algorithm can be used.

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Special Matrices

Square Matrix

Definition

Square Matrix

Let be a matrix. is square if it has the same number of rows and columns.

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Symmetric Matrix

Definition

Symmetric Matrix

A matrix is called symmetric if the tranposed matrix is equal to .

where is the transposition of .

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Invertible Matrix

Definition

Invertible Matrix

A square matrix is called invertible if there exists a matrix such that:

The matrix is called the inverse of . Not invertible matrices are called singular matrices.

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