# Square matrix

In mathematics, 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. Any two square matrices of the same order can be added and multiplied.

Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if is a square matrix representing a rotation and is a column vector describing the position of a point in space, the product yields another column vector describing the position of that point after that rotation. If is a row vector, the same transformation can be obtained using, where is the transpose of.

## Main diagonal

The entries 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. For instance, the main diagonal of the 4-by-4 matrix above contains the elements*a*

_{11}= 9,

*a*

_{22}= 11,

*a*

_{33}= 4,

*a*

_{44}= 10.

The diagonal of a square matrix from the top right to the bottom left corner is called

*antidiagonal*or

*counterdiagonal*.

## Special kinds

### Diagonal or triangular matrix

If all entries outside the main diagonal are zero, is called a diagonal matrix. If only all entries above the main diagonal are zero, ' is called a lower triangular matrix.### Identity matrix

The identity matrix of size is the matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.It is a square matrix of order, and also a special kind of diagonal matrix. It is called identity matrix because multiplication with it leaves a matrix unchanged:

### Symmetric or skew-symmetric matrix

A square matrix**A**that is equal to its transpose, i.e.,, 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, where denotes the conjugate transpose of the matrix, i.e., the transpose of the complex conjugate of.

By the spectral theorem, real symmetric matrices have an orthogonal 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 is called*invertible*or

*non-singular*if there exists a matrix such that

If exists, it is unique and is called the

*inverse matrix*of, denoted.

### Normal matrix

A square matrix**A**is called

*normal*if, i.e. if it commutes with its transpose.

### Definite matrix

A symmetric*n*×

*n*-matrix is called

*positive-definite*, if for all nonzero vectors the associated quadratic form given by

takes only positive values. If the quadratic form takes only non-negative values, the symmetric matrix is called positive-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**:

### Orthogonal matrix

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

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

which entails

where

*I*is the identity matrix.

An orthogonal matrix

*A*is necessarily invertible, unitary, and normal. 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.

## Operations

### Trace

The trace, tr of a square matrix**A**is the sum of its diagonal entries. While matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors:

This is immediate from the definition of matrix multiplication:

Also, the trace of a matrix is equal to that of its transpose, i.e.,

### Determinant

The*determinant*or of a square matrix is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area or volume of the image of the unit square, while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.

The determinant of 2-by-2 matrices is given by

The determinant of 3-by-3 matrices involves 6 terms. The more lengthy Leibniz formula generalises these two formulae to all dimensions.

The determinant of a product of square matrices equals the product of their determinants:

Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices. This expansion can be used for a recursive definition of determinants, that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.

### Eigenvalues and eigenvectors

A number λ and a non-zero vector satisfyingare called an

*eigenvalue*and an

*eigenvector*of, respectively. The number λ is an eigenvalue of an

*n*×

*n*-matrix

**A**if and only if

**A**−λ

**I**

_{n}is not invertible, which is equivalent to

The polynomial

*p*

_{A}in an indeterminate

*X*given by evaluation of the determinant det is called the characteristic polynomial of

**A**. It is a monic polynomial of degree

*n*. Therefore the polynomial equation

*p*

_{A}= 0 has at most

*n*different solutions, i.e., eigenvalues of the matrix. They may be complex even if the entries of

**A**are real. According to the Cayley–Hamilton theorem,

*p*

_{A}=

**0**, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.