Matrices

 

Matrix Notation

Image   Adjoint matrix

Image     Trace of a matrix

Image         A matrix

Image      Hermitian matrix (conjugate transpose)

Image       Transpose matrix

Image       Conjugate matrix

Image      Inverse matrix

Image          Identity matrix

 

Matrix Definitions

Consider a matrix, Image , with complex elements:

 

Image

 

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

 

Adjoint Matrix

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

 

Image

 

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

 

Image

 

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, Image , 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 Image  is commonly denoted as Image .

 

Cofactor Matrix

The matrix of cofactors is found by replacing every element,Image , 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).

 

Image

 

Column Rank

See Rank of a Matrix.

 

Congruent Matrix

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

 

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 Image .

 

Diagonal Matrix

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

 

Image

 

Generalized Inverse Matrix

 

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.

 

Image

 

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 Image .

 

Image

 

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 Image .  Note that a matrix must be square to have an inverse (but see also Generalized Inverse Matrix).

 

Lower Triangular Matrix

See Triangular Matrix.

 

Negative Definite Matrix

 

Negative Semi-Definite 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 Image .

 

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 Image  (read “5 by 3”); and the maximum order is 5.

 

Orthogonal Matrix

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

 

Positive Definite Matrix

 

Positive Semi-definite Matrix

 

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.

 

Image

 

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 Image  for every Image  and Image .  Note that every diagonal element must be 0.

 

Skew-Symmetric Matrix

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

 

Image

 

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 columnsImage .

Image

 

Symmetric Matrix

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

 

Image

 

Trace of a Matrix

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

 

Image

 

the trace is Image .

 

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”.

 

Image

 

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, Image for every Image .  (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, Image for every Image .

 

Upper Triangular:

Image

 

Lower Triangular:

Image

 

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.

 

Image

 

Unitary Matrix

A unitary matrix has the property that Image .

 

Unity Matrix

See Identity Matrix.

 

Upper Triangular Matrix

See Triangular Matrix.

 

Matrix Properties

Image

Image

Image

 

 

If Image  is orthogonal then Image

If Image  is Hermitian then Image

If Image  is skew-Hermitian then Image

If Image  is unitary then Image

If Image  is normal then Image

 

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