# Matrix Notation

Trace of a matrix

A matrix

Hermitian matrix (conjugate transpose)

Transpose matrix

Conjugate matrix

Inverse matrix

Identity matrix

# Matrix Definitions

Consider a matrix, , with complex elements:

Note that each element has the form .  The rows are numbered from 1 to M, and the columns are numbered from 1 to N.

The adjoint of a square matrix is the transpose of the matrix of cofactors.

## Block-Diagonal Matrix

A block-diagonal matrix has square “blocks” of generally non-zero elements situated along its main diagonal with the diagonal of each block coinciding with the diagonal of the matrix.  Elsewhere there are zeroes. Each block must be a square matrix. An example of a block-diagonal matrix is

## Block Matrix

A block matrix is any matrix whose elements have been subdivided into rectangular regions or “blocks.”

## Cofactor

The cofactor of an element of a matrix, say, , is found in the following manner.  First, delete the mth row and nth column of the original square matrix; that is, rewrite the matrix as an M-1 by M-1 matrix omitting the mth row and nth column.  Next, find the determinant of the resulting matrix. The cofactor of element  is commonly denoted as .

## Cofactor Matrix

The matrix of cofactors is found by replacing every element, , of a matrix by its cofactor.

## Conjugate Matrix

All elements of a conjugate matrix are the complex conjugate of the original matrix.  The conjugate matrix is commonly denoted by a superscript “*” (asterisk).

## Column Rank

See Rank of a Matrix.

## Congruent Matrix

A congruent matrix is a matrix, , that has the property that a non-singular matrix, , can be found such that  and  is diagonal and that the number of positive elements in  is invariant of .

## Conjugate Transpose Matrix

See Hermitian Matrix.

## Dense Matrix

A dense matrix has few or no elements equal to 0.  Compare to Sparse Matrix.

## Diagonal

The diagonal (also called the main diagonal) includes all of the elements of a square matrix for which it is true that .

## Diagonal Matrix

A diagonal matrix is a square matrix that has non-zero elements only when .

## Hermitian Matrix

The Hermitian matrix is the conjugate transpose of the original matrix.  The Hermitian is commonly denoted with a superscript “H”. Note that the diagonal elements of a Hermitian matrix must be real.

## Identity Matrix

The identity matrix is a square (diagonal) matrix with all diagonal elements equal to 1 and all non-diagonal elements equal to 0.  It is also called the unity matrix.  The identity matrix is commonly denoted by .

## Inverse Matrix

The inverse matrix is the matrix that, when multiplied by the original matrix, yields the identity matrix.  The inverse is commonly denoted by a superscript “-1”.  That is, the inverse matrix has the property that .  Note that a matrix must be square to have an inverse (but see also Generalized Inverse Matrix).

## Lower Triangular Matrix

See Triangular Matrix.

## Non-Singular Matrix

A non-singular matrix has a rank that is equal to the maximum order of the matrix.

## Normal Matrix

A normal matrix has the property that .

## Order of a Matrix

The order of the matrix is the number of rows or columns of a matrix.  For example, if a matrix has 5 rows and 3 columns, the order is denoted  (read “5 by 3”); and the maximum order is 5.

## Orthogonal Matrix

An orthogonal matrix has the property that .  An orthogonal matrix is always invertible with .

## Rank of a Matrix

The rank of a matrix is the maximum number of linearly independent rows (or columns) of a matrix. The rank is also called the row rank (or column rank).

## Row Rank

See Rank of a Matrix.

## Scalar Matrix

A scalar matrix is a diagonal matrix whose non-zero elements are all equal.  That is, the scalar matrix is a constant times the identity matrix.

## Singular Matrix

A singular matrix has a rank that is less than its maximum order.  For a square matrix, if all elements of a row or column are zero, or if any row (or column) is equal to another row (or column), or if any row (or column) is a linear combination of other rows (or columns), then the matrix is singular.

## Skew-Hermitian Matrix

A skew-Hermitian matrix has the property that  for every  and .  Note that every diagonal element must be 0.

## Skew-Symmetric Matrix

A skew-symmetric matrix is a square matrix that has  for every  and  such that .

## Sparse Matrix

A sparse matrix has many or most of its elements equal to 0.  Compare to Dense Matrix.

## Square Matrix

A square matrix has an equal number of rows and columns .

## Symmetric Matrix

A symmetric matrix is a square matrix that has  for every  and .

## Trace of a Matrix

The sum of the diagonal elements of a square matrix is the trace.  The trace of matrix  is commonly denoted .  For example, for the matrix

the trace is .

## Transpose Matrix

The transpose is formed by switching the rows and columns with each other.  The transpose matrix is commonly denoted with a superscript “T”.

## Triangular Matrix

A triangular matrix is a square matrix that has all of its elements either “below” or “above” the diagonal equal to zero.  An upper triangular matrix has all of its elements “below” the diagonal equal to zero. That is, for every .  (It is an upper triangular matrix because the remaining elements are generally non-zero.)  A lower triangular matrix has all of its elements “above” the diagonal equal to zero. That is, for every .

Upper Triangular:

Lower Triangular:

## Tri-diagonal Matrix

A tri-diagonal matrix has zero elements everywhere except along its main diagonal and the elements immediately adjacent to the main diagonal.  The following is a tri-diagonal matrix.

## Unitary Matrix

A unitary matrix has the property that .

## Unity Matrix

See Identity Matrix.

## Upper Triangular Matrix

See Triangular Matrix.

# Matrix Properties

If  is orthogonal then

If  is Hermitian then

If  is skew-Hermitian then

If  is unitary then

If  is normal then

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