Definite matrix

Not to be confused with Positive matrix and Totally positive matrix.

In mathematics, a symmetric matrix ${\displaystyle M}$ with real entries is positive-definite if the real number ${\displaystyle z^{\operatorname {T} }Mz}$ is positive for every nonzero real column vector ${\displaystyle z,}$ where ${\displaystyle z^{\operatorname {T} }}$ is the transpose of ${\displaystyle z}$.^[1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number ${\displaystyle z^{*}Mz}$ is positive for every nonzero complex column vector ${\displaystyle z,}$ where ${\displaystyle z^{*}}$ denotes the conjugate transpose of ${\displaystyle z.}$

Positive semi-definite matrices are defined similarly, except that the scalars ${\displaystyle z^{\operatorname {T} }Mz}$ and ${\displaystyle z^{*}Mz}$ are required to be positive or zero (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite.

A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines an inner product.

Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathematics. A matrix M is positive-definite if and only if it satisfies any of the following equivalent conditions.

• M is congruent with a diagonal matrix with positive real entries.
• M is symmetric or Hermitian, and all its eigenvalues are real and positive.
• M is symmetric or Hermitian, and all its leading principal minors are positive.
• There exists an invertible matrix ${\displaystyle B}$ with conjugate transpose ${\displaystyle B^{*}}$ such that ${\displaystyle M=B^{*}B.}$

A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed.

Positive-definite and positive-semidefinite real matrices are at the basis of convex optimization, since, given a function of several real variables that is twice differentiable, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a point p, then the function is convex near p, and, conversely, if the function is convex near p, then the Hessian matrix is positive-semidefinite at p.

The set of positive definite matrices is an open convex cone, while the set of positive semi-definite matrices is a closed convex cone.^[2]

Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.

Definitions

In the following definitions, ${\displaystyle \mathbf {x} ^{\operatorname {T} }}$  is the transpose of ${\displaystyle \mathbf {x} }$ , ${\displaystyle \mathbf {x} ^{*}}$  is the conjugate transpose of ${\displaystyle \mathbf {x} }$  and ${\displaystyle \mathbf {0} }$  denotes the n-dimensional zero-vector.

Definitions for real matrices

An ${\displaystyle n\times n}$  symmetric real matrix ${\displaystyle M}$  is said to be positive-definite if ${\displaystyle \mathbf {x} ^{\operatorname {T} }M\mathbf {x} >0}$  for all non-zero ${\displaystyle \mathbf {x} }$  in ${\displaystyle \mathbb {R} ^{n}}$ . Formally,

${\displaystyle M{\text{ positive-definite}}\quad \iff \quad \mathbf {x} ^{\operatorname {T} }M\mathbf {x} >0{\text{ for all }}\mathbf {x} \in \mathbb {R} ^{n}\setminus \{\mathbf {0} \}}$ 

An ${\displaystyle n\times n}$  symmetric real matrix ${\displaystyle M}$  is said to be positive-semidefinite or non-negative-definite if ${\displaystyle \mathbf {x} ^{\operatorname {T} }M\mathbf {x} \geq 0}$  for all ${\displaystyle \mathbf {x} }$  in ${\displaystyle \mathbb {R} ^{n}}$ . Formally,

${\displaystyle M{\text{ positive semi-definite}}\quad \iff \quad \mathbf {x} ^{\operatorname {T} }M\mathbf {x} \geq 0{\text{ for all }}\mathbf {x} \in \mathbb {R} ^{n}}$ 

An ${\displaystyle n\times n}$  symmetric real matrix ${\displaystyle M}$  is said to be negative-definite if ${\displaystyle \mathbf {x} ^{\operatorname {T} }M\mathbf {x} <0}$  for all non-zero ${\displaystyle \mathbf {x} }$  in ${\displaystyle \mathbb {R} ^{n}}$ . Formally,

${\displaystyle M{\text{ negative-definite}}\quad \iff \quad \mathbf {x} ^{\operatorname {T} }M\mathbf {x} <0{\text{ for all }}\mathbf {x} \in \mathbb {R} ^{n}\setminus \{\mathbf {0} \}}$ 

An ${\displaystyle n\times n}$  symmetric real matrix ${\displaystyle M}$  is said to be negative-semidefinite or non-positive-definite if ${\displaystyle x^{\operatorname {T} }Mx\leq 0}$  for all ${\displaystyle x}$  in ${\displaystyle \mathbb {R} ^{n}}$ . Formally,

${\displaystyle M{\text{ negative semi-definite}}\quad \iff \quad \mathbf {x} ^{\operatorname {T} }M\mathbf {x} \leq 0{\text{ for all }}\mathbf {x} \in \mathbb {R} ^{n}}$ 

An ${\displaystyle n\times n}$  symmetric real matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.

Definitions for complex matrices

The following definitions all involve the term ${\displaystyle \mathbf {x} ^{*}M\mathbf {x} }$ . Notice that this is always a real number for any Hermitian square matrix ${\displaystyle M}$ .

An ${\displaystyle n\times n}$  Hermitian complex matrix ${\displaystyle M}$  is said to be positive-definite if ${\displaystyle \mathbf {x} ^{*}M\mathbf {x} >0}$  for all non-zero ${\displaystyle \mathbf {x} }$  in ${\displaystyle \mathbb {C} ^{n}}$ . Formally,

${\displaystyle M{\text{ positive-definite}}\quad \iff \quad \mathbf {x} ^{*}M\mathbf {x} >0{\text{ for all }}\mathbf {x} \in \mathbb {C} ^{n}\setminus \{\mathbf {0} \}}$ 

An ${\displaystyle n\times n}$  Hermitian complex matrix ${\displaystyle M}$  is said to be positive semi-definite or non-negative-definite if ${\displaystyle x^{*}Mx\geq 0}$  for all ${\displaystyle x}$  in ${\displaystyle \mathbb {C} ^{n}}$ . Formally,

${\displaystyle M{\text{ positive semi-definite}}\quad \iff \quad \mathbf {x} ^{*}M\mathbf {x} \geq 0{\text{ for all }}\mathbf {x} \in \mathbb {C} ^{n}}$ 

An ${\displaystyle n\times n}$  Hermitian complex matrix ${\displaystyle M}$  is said to be negative-definite if ${\displaystyle \mathbf {x} ^{*}M\mathbf {x} <0}$  for all non-zero ${\displaystyle \mathbf {x} }$  in ${\displaystyle \mathbb {C} ^{n}}$ . Formally,

${\displaystyle M{\text{ negative-definite}}\quad \iff \quad \mathbf {x} ^{*}M\mathbf {x} <0{\text{ for all }}\mathbf {x} \in \mathbb {C} ^{n}\setminus \{\mathbf {0} \}}$ 

An ${\displaystyle n\times n}$  Hermitian complex matrix ${\displaystyle M}$  is said to be negative semi-definite or non-positive-definite if ${\displaystyle \mathbf {x} ^{*}M\mathbf {x} \leq 0}$  for all ${\displaystyle \mathbf {x} }$  in ${\displaystyle \mathbb {C} ^{n}}$ . Formally,

${\displaystyle M{\text{ negative semi-definite}}\quad \iff \quad \mathbf {x} ^{*}M\mathbf {x} \leq 0{\text{ for all }}\mathbf {x} \in \mathbb {C} ^{n}}$ 

An ${\displaystyle n\times n}$  Hermitian complex matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.

Consistency between real and complex definitions

Since every real matrix is also a complex matrix, the definitions of "definiteness" for the two classes must agree.

For complex matrices, the most common definition says that ${\displaystyle M}$  is positive-definite if and only if ${\displaystyle \mathbf {z} ^{*}M\mathbf {z} }$  is real and positive for every non-zero complex column vectors ${\displaystyle \mathbf {z} }$ . This condition implies that ${\displaystyle M}$  is Hermitian (i.e. its transpose is equal to its conjugate), since ${\displaystyle \mathbf {z} ^{*}M\mathbf {z} }$  being real, it equals its conjugate transpose ${\displaystyle \mathbf {z} ^{*}M^{*}\mathbf {z} }$  for every ${\displaystyle z,}$  which implies ${\displaystyle M=M^{*}}$ .

By this definition, a positive-definite real matrix ${\displaystyle M}$  is Hermitian, hence symmetric; and ${\displaystyle \mathbf {z} ^{\operatorname {T} }M\mathbf {z} }$  is positive for all non-zero real column vectors ${\displaystyle \mathbf {z} }$ . However the last condition alone is not sufficient for ${\displaystyle M}$  to be positive-definite. For example, if

$${\displaystyle M={\begin{bmatrix}1&1\\-1&1\end{bmatrix}},}$$

 

then for any real vector ${\displaystyle \mathbf {z} }$  with entries ${\displaystyle a}$  and ${\displaystyle b}$  we have ${\displaystyle \mathbf {z} ^{\operatorname {T} }M\mathbf {z} =\left(a+b\right)a+\left(-a+b\right)b=a^{2}+b^{2}}$ , which is always positive if ${\displaystyle \mathbf {z} }$  is not zero. However, if ${\displaystyle \mathbf {z} }$  is the complex vector with entries ${\displaystyle 1}$  and ${\displaystyle i}$ , one gets

$${\displaystyle \mathbf {z} ^{*}M\mathbf {z} ={\begin{bmatrix}1&-i\end{bmatrix}}M{\begin{bmatrix}1\\i\end{bmatrix}}={\begin{bmatrix}1+i&1-i\end{bmatrix}}{\begin{bmatrix}1\\i\end{bmatrix}}=2+2i}$$

 

which is not real. Therefore, ${\displaystyle M}$  is not positive-definite.

On the other hand, for a symmetric real matrix ${\displaystyle M}$ , the condition "${\displaystyle \mathbf {z} ^{\operatorname {T} }M\mathbf {z} >0}$  for all nonzero real vectors ${\displaystyle \mathbf {z} }$ " does imply that ${\displaystyle M}$  is positive-definite in the complex sense.

Notation

If a Hermitian matrix ${\displaystyle M}$  is positive semi-definite, one sometimes writes ${\displaystyle M\succeq 0}$  and if ${\displaystyle M}$  is positive-definite one writes ${\displaystyle M\succ 0}$ . To denote that ${\displaystyle M}$  is negative semi-definite one writes ${\displaystyle M\preceq 0}$  and to denote that ${\displaystyle M}$  is negative-definite one writes ${\displaystyle M\prec 0}$ .

The notion comes from functional analysis where positive semidefinite matrices define positive operators. If two matrices ${\displaystyle A}$  and ${\displaystyle B}$  satisfy ${\displaystyle B-A\succeq 0}$ , we can define a non-strict partial order ${\displaystyle B\succeq A}$  that is reflexive, antisymmetric, and transitive; It is not a total order, however, as ${\displaystyle B-A}$  in general may be indefinite.

A common alternative notation is ${\displaystyle M\geq 0}$ , ${\displaystyle M>0}$ , ${\displaystyle M\leq 0}$  and ${\displaystyle M<0}$  for positive semi-definite and positive-definite, negative semi-definite and negative-definite matrices, respectively. This may be confusing, as sometimes nonnegative matrices (respectively, nonpositive matrices) are also denoted in this way.

Examples

• The identity matrix ${\displaystyle I={\begin{bmatrix}1&0\\0&1\end{bmatrix}}}$  is positive-definite (and as such also positive semi-definite). It is a real symmetric matrix, and, for any non-zero column vector z with real entries a and b, one has
  $${\displaystyle \mathbf {z} ^{\operatorname {T} }I\mathbf {z} ={\begin{bmatrix}a&b\end{bmatrix}}{\begin{bmatrix}1&0\\0&1\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}=a^{2}+b^{2}.}$$

   
  Seen as a complex matrix, for any non-zero column vector z with complex entries a and b one has
  $${\displaystyle \mathbf {z} ^{*}I\mathbf {z} ={\begin{bmatrix}{\overline {a}}&{\overline {b}}\end{bmatrix}}{\begin{bmatrix}1&0\\0&1\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}={\overline {a}}a+{\overline {b}}b=|a|^{2}+|b|^{2}.}$$

   
  Either way, the result is positive since ${\displaystyle \mathbf {z} }$  is not the zero vector (that is, at least one of ${\displaystyle a}$  and ${\displaystyle b}$  is not zero).
• The real symmetric matrix
  $${\displaystyle M={\begin{bmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{bmatrix}}}$$

   
  is positive-definite since for any non-zero column vector z with entries a, b and c, we have
  $${\displaystyle {\begin{aligned}\mathbf {z} ^{\operatorname {T} }M\mathbf {z} =\left(\mathbf {z} ^{\operatorname {T} }M\right)\mathbf {z} &={\begin{bmatrix}(2a-b)&(-a+2b-c)&(-b+2c)\end{bmatrix}}{\begin{bmatrix}a\\b\\c\end{bmatrix}}\\&=(2a-b)a+(-a+2b-c)b+(-b+2c)c\\&=2a^{2}-ba-ab+2b^{2}-cb-bc+2c^{2}\\&=2a^{2}-2ab+2b^{2}-2bc+2c^{2}\\&=a^{2}+a^{2}-2ab+b^{2}+b^{2}-2bc+c^{2}+c^{2}\\&=a^{2}+(a-b)^{2}+(b-c)^{2}+c^{2}\end{aligned}}}$$

   
  This result is a sum of squares, and therefore non-negative; and is zero only if ${\displaystyle a=b=c=0}$ , that is, when z is the zero vector.
• For any real invertible matrix ${\displaystyle A}$ , the product ${\displaystyle A^{\operatorname {T} }A}$  is a positive definite matrix (if the means of the columns of A are 0, then this is also called the covariance matrix). A simple proof is that for any non-zero vector ${\displaystyle \mathbf {z} }$ , the condition ${\displaystyle \mathbf {z} ^{\operatorname {T} }A^{\operatorname {T} }A\mathbf {z} =(A\mathbf {z} )^{\operatorname {T} }(A\mathbf {z} )=\|A\mathbf {z} \|^{2}>0,}$  since the invertibility of matrix ${\displaystyle A}$  means that ${\displaystyle A\mathbf {z} \neq 0.}$ 
• The example ${\displaystyle M}$  above shows that a matrix in which some elements are negative may still be positive definite. Conversely, a matrix whose entries are all positive is not necessarily positive definite, as for example
  $${\displaystyle N={\begin{bmatrix}1&2\\2&1\end{bmatrix}},}$$

   
  for which ${\displaystyle {\begin{bmatrix}-1&1\end{bmatrix}}N{\begin{bmatrix}-1&1\end{bmatrix}}^{\operatorname {T} }=-2<0.}$ 

Eigenvalues

Let ${\displaystyle M}$  be an ${\displaystyle n\times n}$  Hermitian matrix (this includes real symmetric matrices). All eigenvalues of ${\displaystyle M}$  are real, and their sign characterize its definiteness:

• ${\displaystyle M}$  is positive definite if and only if all of its eigenvalues are positive.
• ${\displaystyle M}$  is positive semi-definite if and only if all of its eigenvalues are non-negative.
• ${\displaystyle M}$  is negative definite if and only if all of its eigenvalues are negative
• ${\displaystyle M}$  is negative semi-definite if and only if all of its eigenvalues are non-positive.
• ${\displaystyle M}$  is indefinite if and only if it has both positive and negative eigenvalues.

Let ${\displaystyle PDP^{-1}}$  be an eigendecomposition of ${\displaystyle M}$ , where ${\displaystyle P}$  is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors of ${\displaystyle M}$ , and ${\displaystyle D}$  is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues. The matrix ${\displaystyle M}$  may be regarded as a diagonal matrix ${\displaystyle D}$  that has been re-expressed in coordinates of the (eigenvectors) basis ${\displaystyle P}$ . Put differently, applying ${\displaystyle M}$  to some vector z, giving Mz, is the same as changing the basis to the eigenvector coordinate system using P⁻¹, giving P⁻¹z, applying the stretching transformation D to the result, giving DP⁻¹z, and then changing the basis back using P, giving PDP⁻¹z.

With this in mind, the one-to-one change of variable ${\displaystyle \mathbf {y} =P\mathbf {z} }$  shows that ${\displaystyle \mathbf {z} ^{*}M\mathbf {z} }$  is real and positive for any complex vector ${\displaystyle \mathbf {z} }$  if and only if ${\displaystyle \mathbf {y} ^{*}D\mathbf {y} }$  is real and positive for any ${\displaystyle y}$ ; in other words, if ${\displaystyle D}$  is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal—that is, every eigenvalue of ${\displaystyle M}$ —is positive. Since the spectral theorem guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using Descartes' rule of alternating signs when the characteristic polynomial of a real, symmetric matrix ${\displaystyle M}$  is available.

Decomposition

See also: Gram matrix

Let ${\displaystyle M}$  be an ${\displaystyle n\times n}$  Hermitian matrix. ${\displaystyle M}$  is positive semidefinite if and only if it can be decomposed as a product

$${\displaystyle M=B^{*}B}$$

 

of a matrix ${\displaystyle B}$  with its conjugate transpose.

When ${\displaystyle M}$  is real, ${\displaystyle B}$  can be real as well and the decomposition can be written as

$${\displaystyle M=B^{\operatorname {T} }B.}$$

 

${\displaystyle M}$  is positive definite if and only if such a decomposition exists with ${\displaystyle B}$  invertible. More generally, ${\displaystyle M}$  is positive semidefinite with rank ${\displaystyle k}$  if and only if a decomposition exists with a ${\displaystyle k\times n}$  matrix ${\displaystyle B}$  of full row rank (i.e. of rank ${\displaystyle k}$ ). Moreover, for any decomposition ${\displaystyle M=B^{*}B}$ , ${\displaystyle \operatorname {rank} (M)=\operatorname {rank} (B)}$ .^[3]

Proof

If ${\displaystyle M=B^{*}B}$ , then ${\displaystyle x^{*}Mx=(x^{*}B^{*})(Bx)=\|Bx\|^{2}\geq 0}$ , so ${\displaystyle M}$  is positive semidefinite. If moreover ${\displaystyle B}$  is invertible then the inequality is strict for ${\displaystyle x\neq 0}$ , so ${\displaystyle M}$  is positive definite. If ${\displaystyle B}$  is ${\displaystyle k\times n}$  of rank ${\displaystyle k}$ , then ${\displaystyle \operatorname {rank} (M)=\operatorname {rank} (B^{*})=k}$ .

In the other direction, suppose ${\displaystyle M}$  is positive semidefinite. Since ${\displaystyle M}$  is Hermitian, it has an eigendecomposition ${\displaystyle M=Q^{-1}DQ}$  where ${\displaystyle Q}$  is unitary and ${\displaystyle D}$  is a diagonal matrix whose entries are the eigenvalues of ${\displaystyle M}$  Since ${\displaystyle M}$  is positive semidefinite, the eigenvalues are non-negative real numbers, so one can define ${\displaystyle D^{\frac {1}{2}}}$  as the diagonal matrix whose entries are non-negative square roots of eigenvalues. Then ${\displaystyle M=Q^{-1}DQ=Q^{*}DQ=Q^{*}D^{\frac {1}{2}}D^{\frac {1}{2}}Q=Q^{*}D^{{\frac {1}{2}}*}D^{\frac {1}{2}}Q=B^{*}B}$  for ${\displaystyle B=D^{\frac {1}{2}}Q}$ . If moreover ${\displaystyle M}$  is positive definite, then the eigenvalues are (strictly) positive, so ${\displaystyle D^{\frac {1}{2}}}$  is invertible, and hence ${\displaystyle B=D^{\frac {1}{2}}Q}$  is invertible as well. If ${\displaystyle M}$  has rank ${\displaystyle k}$ , then it has exactly ${\displaystyle k}$  positive eigenvalues and the others are zero, hence in ${\displaystyle B=D^{\frac {1}{2}}Q}$  all but ${\displaystyle k}$  rows are all zeroed. Cutting the zero rows gives a ${\displaystyle k\times n}$  matrix ${\displaystyle B'}$  such that ${\displaystyle B'^{*}B'=B^{*}B=M}$ .

The columns ${\displaystyle b_{1},\dots ,b_{n}}$  of ${\displaystyle B}$  can be seen as vectors in the complex or real vector space ${\displaystyle \mathbb {R} ^{k}}$ , respectively. Then the entries of ${\displaystyle M}$  are inner products (that is dot products, in the real case) of these vectors

$${\displaystyle M_{ij}=\langle b_{i},b_{j}\rangle .}$$

 

In other words, a Hermitian matrix ${\displaystyle M}$  is positive semidefinite if and only if it is the Gram matrix of some vectors ${\displaystyle b_{1},\dots ,b_{n}}$ . It is positive definite if and only if it is the Gram matrix of some linearly independent vectors. In general, the rank of the Gram matrix of vectors ${\displaystyle b_{1},\dots ,b_{n}}$  equals the dimension of the space spanned by these vectors.^[4]

Uniqueness up to unitary transformations

The decomposition is not unique: if ${\displaystyle M=B^{*}B}$  for some ${\displaystyle k\times n}$  matrix ${\displaystyle B}$  and if ${\displaystyle Q}$  is any unitary ${\displaystyle k\times k}$  matrix (meaning ${\displaystyle Q^{*}Q=QQ^{*}=I}$ ), then ${\displaystyle M=B^{*}B=B^{*}Q^{*}QB=A^{*}A}$  for ${\displaystyle A=QB}$ .

However, this is the only way in which two decompositions can differ: the decomposition is unique up to unitary transformations. More formally, if ${\displaystyle A}$  is a ${\displaystyle k\times n}$  matrix and ${\displaystyle B}$  is a ${\displaystyle \ell \times n}$  matrix such that ${\displaystyle A^{*}A=B^{*}B}$ , then there is a ${\displaystyle \ell \times k}$  matrix ${\displaystyle Q}$  with orthonormal columns (meaning ${\displaystyle Q^{*}Q=I_{k\times k}}$ ) such that ${\displaystyle B=QA}$ .^[5] When ${\displaystyle \ell =k}$  this means ${\displaystyle Q}$  is unitary.

This statement has an intuitive geometric interpretation in the real case: let the columns of ${\displaystyle A}$  and ${\displaystyle B}$  be the vectors ${\displaystyle a_{1},\dots ,a_{n}}$  and ${\displaystyle b_{1},\dots ,b_{n}}$  in ${\displaystyle \mathbb {R} ^{k}}$ . A real unitary matrix is an orthogonal matrix, which describes a rigid transformation (an isometry of Euclidean space ${\displaystyle \mathbb {R} ^{k}}$ ) preserving the 0 point (i.e. rotations and reflections, without translations). Therefore, the dot products ${\displaystyle a_{i}\cdot a_{j}}$  and ${\displaystyle b_{i}\cdot b_{j}}$  are equal if and only if some rigid transformation of ${\displaystyle \mathbb {R} ^{k}}$  transforms the vectors ${\displaystyle a_{1},\dots ,a_{n}}$  to ${\displaystyle b_{1},\dots ,b_{n}}$  (and 0 to 0).

Square root

Main article: Square root of a matrix

A Hermitian matrix ${\displaystyle M}$  is positive semidefinite if and only if there is a positive semidefinite matrix ${\displaystyle B}$  (in particular ${\displaystyle B}$  is Hermitian, so ${\displaystyle B^{*}=B}$ ) satisfying ${\displaystyle M=BB}$ . This matrix ${\displaystyle B}$  is unique,^[6] is called the non-negative square root of ${\displaystyle M}$ , and is denoted with ${\displaystyle B=M^{\frac {1}{2}}}$ . When ${\displaystyle M}$  is positive definite, so is ${\displaystyle M^{\frac {1}{2}}}$ , hence it is also called the positive square root of ${\displaystyle M}$ .

The non-negative square root should not be confused with other decompositions ${\displaystyle M=B^{*}B}$ . Some authors use the name square root and ${\displaystyle M^{\frac {1}{2}}}$  for any such decomposition, or specifically for the Cholesky decomposition, or any decomposition of the form ${\displaystyle M=BB}$ ; others only use it for the non-negative square root.

If ${\displaystyle M>N>0}$  then ${\displaystyle M^{\frac {1}{2}}>N^{\frac {1}{2}}>0}$ .

Cholesky decomposition

A Hermitian positive semidefinite matrix ${\displaystyle M}$  can be written as ${\displaystyle M=LL^{*}}$ , where ${\displaystyle L}$  is lower triangular with non-negative diagonal (equivalently ${\displaystyle M=B^{*}B}$  where ${\displaystyle B=L^{*}}$  is upper triangular); this is the Cholesky decomposition. If ${\displaystyle M}$  is positive definite, then the diagonal of ${\displaystyle L}$  is positive and the Cholesky decomposition is unique. Conversely if ${\displaystyle L}$  is lower triangular with nonnegative diagonal then ${\displaystyle LL^{*}}$  is positive semidefinite. The Cholesky decomposition is especially useful for efficient numerical calculations. A closely related decomposition is the LDL decomposition, ${\displaystyle M=LDL^{*}}$ , where ${\displaystyle D}$  is diagonal and ${\displaystyle L}$  is lower unitriangular.

Other characterizations

Let ${\displaystyle M}$  be an ${\displaystyle n\times n}$  real symmetric matrix, and let ${\displaystyle B_{1}(M):=\{x\in \mathbb {R} ^{n}:x^{\operatorname {T} }Mx\leq 1\}}$  be the "unit ball" defined by ${\displaystyle M}$ . Then we have the following

• ${\displaystyle B_{1}(vv^{\operatorname {T} })}$  is a solid slab sandwiched between ${\displaystyle \pm \{w:\langle w,v\rangle =1\}}$ .
• ${\displaystyle M\succeq 0}$  if and only if ${\displaystyle B_{1}(M)}$  is an ellipsoid, or an ellipsoidal cylinder.
• ${\displaystyle M\succ 0}$  if and only if ${\displaystyle B_{1}(M)}$  is bounded, that is, it is an ellipsoid.
• If ${\displaystyle N\succ 0}$ , then ${\displaystyle M\succeq N}$  if and only if ${\displaystyle B_{1}(M)\subseteq B_{1}(N)}$ ; ${\displaystyle M\succ N}$  if and only if ${\displaystyle B_{1}(M)\subseteq \operatorname {int} (B_{1}(N))}$ .
• If ${\displaystyle N\succ 0}$  , then ${\displaystyle M\succeq {\frac {vv^{\operatorname {T} }}{v^{\operatorname {T} }Nv}}}$  for all ${\displaystyle v\neq 0}$  if and only if ${\textstyle B_{1}(M)\subset \bigcap _{v^{\operatorname {T} }Nv=1}B_{1}(vv^{\operatorname {T} })}$ . So, since the polar dual of an ellipsoid is also an ellipsoid with the same principal axes, with inverse lengths, we have
  $${\displaystyle B_{1}(N^{-1})=\bigcap _{v^{\operatorname {T} }Nv=1}B_{1}(vv^{\operatorname {T} })=\bigcap _{v^{\operatorname {T} }Nv=1}\{w:|\langle w,v\rangle |\leq 1\}}$$

   
  That is, if ${\displaystyle N}$  is positive-definite, then ${\displaystyle M\succeq {\frac {vv^{\operatorname {T} }}{v^{\operatorname {T} }Nv}}}$  for all ${\displaystyle v\neq 0}$  if and only if ${\displaystyle M\succeq N^{-1}}$ 

Let ${\displaystyle M}$  be an ${\displaystyle n\times n}$  Hermitian matrix. The following properties are equivalent to ${\displaystyle M}$  being positive definite:

The associated sesquilinear form is an inner product

The sesquilinear form defined by ${\displaystyle M}$  is the function ${\displaystyle \langle \cdot ,\cdot \rangle }$  from ${\displaystyle \mathbb {C} ^{n}\times \mathbb {C} ^{n}}$  to ${\displaystyle \mathbb {C} ^{n}}$  such that ${\displaystyle \langle x,y\rangle :=y^{*}Mx}$  for all ${\displaystyle x}$  and ${\displaystyle y}$  in ${\displaystyle \mathbb {C} ^{n}}$ , where ${\displaystyle y^{*}}$  is the conjugate transpose of ${\displaystyle y}$ . For any complex matrix ${\displaystyle M}$ , this form is linear in ${\displaystyle x}$  and semilinear in ${\displaystyle y}$ . Therefore, the form is an inner product on ${\displaystyle \mathbb {C} ^{n}}$  if and only if ${\displaystyle \langle z,z\rangle }$  is real and positive for all nonzero ${\displaystyle z}$ ; that is if and only if ${\displaystyle M}$  is positive definite. (In fact, every inner product on ${\displaystyle \mathbb {C} ^{n}}$  arises in this fashion from a Hermitian positive definite matrix.)

Its leading principal minors are all positive

The kth leading principal minor of a matrix ${\displaystyle M}$  is the determinant of its upper-left ${\displaystyle k\times k}$  sub-matrix. It turns out that a matrix is positive definite if and only if all these determinants are positive. This condition is known as Sylvester's criterion, and provides an efficient test of positive definiteness of a symmetric real matrix. Namely, the matrix is reduced to an upper triangular matrix by using elementary row operations, as in the first part of the Gaussian elimination method, taking care to preserve the sign of its determinant during pivoting process. Since the kth leading principal minor of a triangular matrix is the product of its diagonal elements up to row ${\displaystyle k}$ , Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive. This condition can be checked each time a new row ${\displaystyle k}$  of the triangular matrix is obtained.

A positive semidefinite matrix is positive definite if and only if it is invertible.^[7] A matrix ${\displaystyle M}$  is negative (semi)definite if and only if ${\displaystyle -M}$  is positive (semi)definite.

Quadratic forms

Main article: Definite quadratic form

The (purely) quadratic form associated with a real ${\displaystyle n\times n}$  matrix ${\displaystyle M}$  is the function ${\displaystyle Q:\mathbb {R} ^{n}\to \mathbb {R} }$  such that ${\displaystyle Q(x)=x^{\operatorname {T} }Mx}$  for all ${\displaystyle x}$ . ${\displaystyle M}$  can be assumed symmetric by replacing it with ${\displaystyle {\tfrac {1}{2}}\left(M+M^{\operatorname {T} }\right)}$ .

A symmetric matrix ${\displaystyle M}$  is positive definite if and only if its quadratic form is a strictly convex function.

More generally, any quadratic function from ${\displaystyle \mathbb {R} ^{n}}$  to ${\displaystyle \mathbb {R} }$  can be written as ${\displaystyle x^{\operatorname {T} }Mx+x^{\operatorname {T} }b+c}$  where ${\displaystyle M}$  is a symmetric ${\displaystyle n\times n}$  matrix, ${\displaystyle b}$  is a real ${\displaystyle n}$ -vector, and ${\displaystyle c}$  a real constant. In the ${\displaystyle n=1}$  case, this is a parabola, and just like in the ${\displaystyle n=1}$  case, we have

Theorem: This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only if ${\displaystyle M}$  is positive definite.

Proof: If ${\displaystyle M}$  is positive definite, then the function is strictly convex. Its gradient is zero at the unique point of ${\displaystyle M^{-1}b}$ , which must be the global minimum since the function is strictly convex. If ${\displaystyle M}$  is not positive definite, then there exists some vector ${\displaystyle v}$  such that ${\displaystyle v^{\operatorname {T} }Mv\leq 0}$ , so the function ${\displaystyle f(t):=(vt)^{\operatorname {T} }M(vt)+b^{\operatorname {T} }(vt)+c}$  is a line or a downward parabola, thus not strictly convex and not having a global minimum.

For this reason, positive definite matrices play an important role in optimization problems.

Simultaneous diagonalization

One symmetric matrix and another matrix that is both symmetric and positive definite can be simultaneously diagonalized. This is so although simultaneous diagonalization is not necessarily performed with a similarity transformation. This result does not extend to the case of three or more matrices. In this section we write for the real case. Extension to the complex case is immediate.

Let ${\displaystyle M}$  be a symmetric and ${\displaystyle N}$  a symmetric and positive definite matrix. Write the generalized eigenvalue equation as ${\displaystyle \left(M-\lambda N\right)\mathbf {x} =0}$  where we impose that ${\displaystyle x}$  be normalized, i.e. ${\displaystyle \mathbf {x} ^{\operatorname {T} }N\mathbf {x} =1}$ . Now we use Cholesky decomposition to write the inverse of ${\displaystyle N}$  as ${\displaystyle Q^{\operatorname {T} }Q}$ . Multiplying by ${\displaystyle Q}$  and letting ${\displaystyle \mathbf {x} =Q^{\operatorname {T} }\mathbf {y} }$ , we get ${\displaystyle Q\left(M-\lambda N\right)Q^{\operatorname {T} }\mathbf {y} =0}$ , which can be rewritten as ${\displaystyle \left(QMQ^{\operatorname {T} }\right)\mathbf {y} =\lambda \mathbf {y} }$  where ${\displaystyle \mathbf {y} ^{\operatorname {T} }\mathbf {y} =1}$ . Manipulation now yields ${\displaystyle MX=NX\Lambda }$  where ${\displaystyle X}$  is a matrix having as columns the generalized eigenvectors and ${\displaystyle \Lambda }$  is a diagonal matrix of the generalized eigenvalues. Now premultiplication with ${\displaystyle X^{\operatorname {T} }}$  gives the final result: ${\displaystyle X^{\operatorname {T} }MX=\Lambda }$  and ${\displaystyle X^{\operatorname {T} }NX=I}$ , but note that this is no longer an orthogonal diagonalization with respect to the inner product where ${\displaystyle \mathbf {y} ^{\operatorname {T} }\mathbf {y} =1}$ . In fact, we diagonalized ${\displaystyle M}$  with respect to the inner product induced by ${\displaystyle N}$ .^[8]

Note that this result does not contradict what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation. Our result here is more akin to a simultaneous diagonalization of two quadratic forms, and is useful for optimization of one form under conditions on the other.

Properties

Induced partial ordering

For arbitrary square matrices ${\displaystyle M}$ , ${\displaystyle N}$  we write ${\displaystyle M\geq N}$  if ${\displaystyle M-N\geq 0}$  i.e., ${\displaystyle M-N}$  is positive semi-definite. This defines a partial ordering on the set of all square matrices. One can similarly define a strict partial ordering ${\displaystyle M>N}$ . The ordering is called the Loewner order.

Inverse of positive definite matrix

Every positive definite matrix is invertible and its inverse is also positive definite.^[9] If ${\displaystyle M\geq N>0}$  then ${\displaystyle N^{-1}\geq M^{-1}>0}$ .^[10] Moreover, by the min-max theorem, the kth largest eigenvalue of ${\displaystyle M}$  is greater than or equal to the kth largest eigenvalue of ${\displaystyle N}$ .

Scaling

If ${\displaystyle M}$  is positive definite and ${\displaystyle r>0}$  is a real number, then ${\displaystyle rM}$  is positive definite.^[11]

Addition

• If ${\displaystyle M}$  and ${\displaystyle N}$  are positive-definite, then the sum ${\displaystyle M+N}$  is also positive-definite.^[11]
• If ${\displaystyle M}$  and ${\displaystyle N}$  are positive-semidefinite, then the sum ${\displaystyle M+N}$  is also positive-semidefinite.
• If ${\displaystyle M}$  is positive-definite and ${\displaystyle N}$  is positive-semidefinite, then the sum ${\displaystyle M+N}$  is also positive-definite.

Multiplication

• If ${\displaystyle M}$  and ${\displaystyle N}$  are positive definite, then the products ${\displaystyle MNM}$  and ${\displaystyle NMN}$  are also positive definite. If ${\displaystyle MN=NM}$ , then ${\displaystyle MN}$  is also positive definite.
• If ${\displaystyle M}$  is positive semidefinite, then ${\displaystyle A^{*}MA}$  is positive semidefinite for any (possibly rectangular) matrix ${\displaystyle A}$ . If ${\displaystyle M}$  is positive definite and

January 01, 1970