### Square Matrices in Mathematics

## Square matrices

A square matrix is a matrix with the same number of rows and columns. An*n*-by-

*n*matrix is known as a square matrix of order

*n.*Any two square matrices of the same order can be added and multiplied. The entries

*a*

_{ii}form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix.

### Main types

Name Example with *n*= 3Diagonal matrix Lower triangular matrix Upper triangular matrix

#### Diagonal or triangular matrix

If all entries outside the main diagonal are zero,**A**is called a diagonal matrix. If only all entries above (or below) the main diagonal are zero,

**A**is called a lower (or upper) triangular matrix.

#### Identity matrix

The identity matrix**I**

_{n}of size

*n*is the

*n*-by-

*n*matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.

*n*, and also a special kind of diagonal matrix. It is called identity matrix because multiplication with it leaves a matrix unchanged:

**AI**_{n}=**I**_{m}**A**=**A**for any*m*-by-*n*matrix**A**.

#### Symmetric or skew-symmetric matrix

A square matrix**A**that is equal to its transpose, i.e.,

**A**=

**A**

^{T}, is a symmetric matrix. If instead,

**A**was equal to the negative of its transpose, i.e.,

**A**= −

**A**

^{T}, then

**A**is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy

**A**

^{∗}=

**A**, where the star or asterisk denotes the conjugate transpose of the matrix, i.e., the transpose of the complex conjugate of

**A**.

By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.

^{}This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below.

#### Invertible matrix and its inverse

A square matrix**A**is called

*invertible*or

*non-singular*if there exists a matrix

**B**such that

**AB**=**BA**=**I**_{n}.^{}^{}

**B**exists, it is unique and is called the

*inverse matrix*of

**A**, denoted

**A**

^{−1}.

#### Definite matrix

Positive definite matrix | Indefinite matrix |
---|---|

Q(x,y) = 1/4 x^{2} + y^{2} |
Q(x,y) = 1/4 x^{2} − 1/4 y^{2} |

Points such that Q(x,y)=1(Ellipse). |
Points such that Q(x,y)=1(Hyperbola). |

*n*×

*n*-matrix is called

*positive-definite*(respectively negative-definite; indefinite), if for all nonzero vectors

**x**∈

**R**

^{n}the associated quadratic form given by

*Q*(**x**) =**x**^{T}**Ax**

^{}If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive.

^{}The table at the right shows two possibilities for 2-by-2 matrices.

Allowing as input two different vectors instead yields the bilinear form associated to

**A**:

*B*_{A}(**x**,**y**) =**x**^{T}**Ay**.^{}

#### Orthogonal matrix

An*orthogonal matrix*is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix

*A*is orthogonal if its transpose is equal to its inverse:

*I*is the identity matrix.

An orthogonal matrix

*A*is necessarily invertible (with inverse

*A*

^{−1}=

*A*

^{T}), unitary (

*A*

^{−1}=

*A**), and normal (

*A**

*A*=

*AA**). The determinant of any orthogonal matrix is either +1 or −1. A

*special orthogonal matrix*is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant +1 is a pure rotation, while every orthogonal matrix with determinant -1 is either a pure reflection, or a composition of reflection and rotation.

The complex analogue of an orthogonal matrix is a unitary matrix.

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