# Properties of a matrix

## Singular matrix

A matrix is singular if and only if its determinant is zero.

Another equivalent definition is that a matrix is singular when at least one column (or row) of the matrix is a linear combination of other columns (or rows) of that matrix. In this case, we say that the matrix columns (or rows) are collinear.

A singular matrix is also called to be **degenerate** or **non invertible**.

**nonsingular**or

**regular**or

**invertible**or

**nondegenerate**, i.e. a matrix whose determinant is non-zero or, equivalently, which has linearly independent columns (and rows). Example :

`M = [[1,2,5],[-8,6,0],[2,4,10]]`

We notice that the 3d row can be obtained by multiplying the 1st row by 2. Since the row vectors of this matrix are not linearly independent, this matrix is singular. All these propositions are equivalent and true for square matrix M,

- Row vectors (or column vectors) are collinear

- Row vectors (or column vectors) are linearly dependent

- Row vectors (or column vectors) are not linearly independent

- The determinant is zero

- This matrix is singular

- This matrix is not regular

- This matrix is not invertible

- This matrix is degenerate

Therefore, to prove that a matrix is singular, just prove that a column (or a row) is a linear combination of other columns (or rows). If such a linear combination is not easy to find then calculate its determinant. If it is zero then the matrix is singular.

## Invertible matrix

A matrix M of size n × n is invertible if there exists a matrix denoted `M^(-1)` of size n × n such as,

`M*M^(-1) = M^(-1) * M = I_n`

I_n is the identity matrix of size n &mult; n.

"M is invertible" is equivalent to "M is not singular".

## Diagonalizable matrix

A square matrix M is diagonalizable if there exists an invertible matrix P and a diagonal matrix D such that,

`A = P D P^(-1)`

In practice, here are the steps to follow to diagonalize a matrix n-by-n.

First, we calculate its characteristic polynomial.

Then, we calculate its eigenvectors, eigenvalues and their multiplicities. As a remainder, eigenvalues are the roots of the matrix characteristic polynomial and the eigenvalues multiplicities are the dimensions of the matrix eigenspaces.

If the the sum of the eigenspaces dimensions is equal to n, then matrix M is diagonalizable. In particular, when M has n distinct eigenvalues then all eigenspaces are of dimension 1 and the matrix is diagonalizable.

## Positive definite matrix

A square matrix M with real entries is **positive-definite** if it satisfies all of these conditions.

- M is a symmetric matrix.

- M is invertible.

- all M eigenvalues are real and positive.

## Negative definite matrix

A square matrix with real entries is **negative-definite** if its additive inverse matrix (-1) * M is positive negative.

## Orthogonal matrix

Let M be a square matrix of size n then M is called orthogonal if it satisfies one of the two following equivalent propostions :

1/ \(M^T . M = M . M^T = I_n\)

where, \(I_n\) is the identity matrix of order n and,

\(M^T\) is the transpose matrix of M.

2/ M is invertible and \(M^{-1} = M^T\)

Example:

`M = [[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]`

It is easy to check that,

\( M^T . M = M .M^T = I_2\)

because of,

`cos^2(theta)+sin^2(theta)= 1`

## Symetric matrix

A square matrix M is symmetric if it is equal to its own transpose matrix i.e.,

\({}^t \! M = M\)

which is equivalent to,

if `a_(ij)` are the entries of matrix M, then,

`a_(ij) = a_(ji)` for all i and j between 1 and n.

## Square matrix

A matrix is square if it has the same number of lines and columns. Square matrice M is said of order n if it is n-by-n.

## Unitary Matrix

A square matrix M with complex coefficients is unitary if it satisfies the following equalities,

\(M . M^{*} = M^{*} . M = I_n\)

- Matrix M is n-by-n.

- \(M^{*}\) is the conjugate transpose of M

- \(I_n\) is the identity matrix of order n

This is equivalent to the fact that M is invertible and its inverse is its own conjugate transpose matrix.

For a matrix with real entries, this is equivalent to say that "M is an orthogonal matrix"

## Normal matrix

A sqaure matrix with complex coefficients is normal if it commutes with its conjugate transpose, i.e.,

\(M . M^{*} = M^{*} . M\)

## Involutory Matrix

An involutory matrix is an invertible square matrix that is equal to its own inverse matrix. Therefore, if M is a square matrix of size n, then it is involutory if and only if,

\(M^{-1} = M\)

which is equivalent to,
\(M^{2} = I_n\)

\(I_n\) being the identity matrix of order n.

## Hermitian matrix

A matrix with complex coefficients is hermitian if it is equal to its own conjugate transpose matrix, that is,

\(M = M^{*}\)

## Nilpotent matrix

A matrix is nilpotent if it exists a power of this matrix which is equal to the zero matrix, i.e.,

It exists a positive integer n such that,

`M^n = 0`

If n is the least positive integer that satisfies this equality, then M is nilpotent of index n.

## Diagonal matrix

A diagonal matrix is a matrix in which all elements outside the principal diagonal are equal to zero.

In a space of dimension n, it can be written as,

\(D_n = \begin{pmatrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_n \\ \end{pmatrix}\)

Example in dimension 2 :

\(D_2 = \begin{pmatrix} 3 & 0 \\ 0 & 4 \\ \end{pmatrix}\)

Example in dimension 3 :

\(D_3 = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix}\)

## See also

Matrix Operations

Determinant of a matrix

conjugate transpose

Inverse matrix

Characteristic Polynomial

Eigenvalues and eigenvectors