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- Definite matrix

*M*

*z*^{sf{T}Mz}

*z,*

*z*^{sf{T}}

*z*^{*}*Mz*

*z,*

*z*^{*}

*z.*

**Positive semi-definite** matrices are defined similarly, except that the scalars

*z*^{sf{T}Mz}

*z*^{*}*Mz*

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 is positive-definite if and only if it satisfies any of the following equivalent conditions.

- is congruent with a diagonal matrix with positive real entries.
- is symmetric or Hermitian, and all its eigenvalues are real and positive .
- is symmetric or Hermitian, and all its leading principal minors are positive.

*B*

*B*^{*}

*M*=*B*^{*B.}

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, then the function is convex near, and, conversely, if the function is convex near, then the Hessian matrix is positive-semidefinite at .

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

In the following definitions,

x^{sf{T}}

x

x^{*}

x

0

An

*n* x *n*

*M*

x^{sf{T}Mx}*>*0

x

*\R*^{n}

An

*n* x *n*

*M*

x^{sf{T}Mx}*\geq*0

x

*\R*^{n}

An

*n* x *n*

*M*

x^{sf{T}Mx}*<*0

x

*\R*^{n}

An

*n* x *n*

*M*

*x*^{sf{T}Mx}*\leq*0

*x*

*\R*^{n}

An

*n* x *n*

The following definitions all involve the term

x^{*}*Mx*

*M*

An

*n* x *n*

*M*

x^{*}*Mx**>*0

x

*\C*^{n}

An

*n* x *n*

*M*

*x*^{*}*Mx**\geq*0

*x*

C^{n}

An

*n* x *n*

*M*

x^{*}*Mx**<*0

x

*\C*^{n}

An

*n* x *n*

*M*

x^{*}*Mx**\leq*0

x

*\C*^{n}

An

*n* x *n*

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 "

*M*

z^{*}*Mz*

z

*M*

*M*=*A*+*iB*

z^{*}*Mz*=z^{*}*Az*+*iz*^{*}*Bz*

*A*

*B*

z^{*}*Az*

z^{*}*Bz*

z^{*}*Mz*

z^{*}*Bz*

z

*B*

*M*=*A*

*M*

By this definition, a positive-definite *real* matrix

*M*

z^{sf{T}Mz}

z

*M*

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

then for any real vector

z

*a*

*b*

z^{sf{T}Mz}=*\left(a*+*b\right)a*+*\left(*-*a*+*b\right)b*=*a*^{2}+*b*^{2}

z

z

1

*i*

z^{*}*M*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+2*i*

which is not real. Therefore,

*M*

On the other hand, for a *symmetric* real matrix

*M*

z^{sf{T}Mz}*>*0

z

*M*

If a Hermitian matrix

*M*

*M**\succeq*0

*M*

*M**\succ*0

*M*

*M**\preceq*0

*M*

*M**\prec*0

The notion comes from functional analysis where positive semidefinite matrices define positive operators.

A common alternative notation is

*M**\geq*0

*M**>*0

*M**\leq*0

*M**<*0

Let

*M*

*n* x *n*

*M*

*M*

*M*

*M*

*M*

*M*

Let

*PD**P*^{-1}

*M*

*P*

*M*

*D*

*M*

*D*

*P*

With this in mind, the one-to-one change of variable

y=*Pz*

z^{*}*Mz*

z

y^{*}*Dy*

*y*

*D*

*M*

*M*

See also: Gram matrix. Let

*M*

*n* x *n*

*M*

*M*=*B*^{*}*B*

*B*

When

*M*

*B*

*M*=*B*^{sf{T}B.}

*M*

*B*

*M*

*k*

*k* x *n*

*B*

*k*

*M*=*B*^{*}*B*

*\operatorname{rank}(M)*=*\operatorname{rank}(B)*

If

*M*=*B*^{*}*B*

*x*^{*}*M**x*=*(x*^{*}*B*^{*)}*(B**x)*=*\|B**x**\|*^{2}*\geq*0

*M*

*B*

*x* ≠ 0

*M*

*B*

*k* x *n*

*k*

*\operatorname{rank}(M)*=*\operatorname{rank}(B*^{*)}=*k*

In the other direction, suppose

*M*

*M*

*M*=*Q*^{-1}*D**Q*

*Q*

*D*

*M*

*M*

| ||||

D |

*M*=*Q*^{-1}*D**Q*=*Q*^{*}*D**Q*=*Q*^{*}

| ||||

D |

| ||||

D |

*Q*=*Q*^{*}

| |||||

D |

| ||||

D |

*Q*=*B*^{*}*B*

*B*=

| ||||

D |

*Q*

*M*

| ||||

D |

*B*=

| ||||

D |

*Q*

*M*

*k*

*k*

*B*=

| ||||

D |

*Q*

*k*

*k* x *n*

*B'*

*B'*^{*}*B'*=*B*^{*}*B*=*M*

The columns

*b*_{1,...,b}_{n}

*B*

R^{k}

*M*

*M*_{ij}=*\langle**b*_{i,}*b*_{j\rangle.}

*M*

*b*_{1,...,b}_{n}

*b*_{1,...,b}_{n}

The decomposition is not unique: if

*M*=*B*^{*}*B*

*k* x *n*

*B*

*Q*

*k* x *k*

*Q*^{*}*Q*=*Q**Q*^{*}=*I*

*M*=*B*^{*}*B*=*B*^{*}*Q*^{*}*Q**B*=*A*^{*}*A*

*A*=*Q**B*

However, this is the only way in which two decompositions can differ: the decomposition is unique up to unitary transformations.More formally, if

*A*

*k* x *n*

*B*

*\ell* x *n*

*A*^{*}*A*=*B*^{*}*B*

*\ell* x *k*

*Q*

*Q*^{*}*Q*=*I*_{k}

*B*=*Q**A*

*\ell*=*k*

*Q*

This statement has an intuitive geometric interpretation in the real case:let the columns of

*A*

*B*

*a*_{1,...,a}_{n}

*b*_{1,...,b}_{n}

R^{k}

R^{k}

*a*_{i} ⋅ *a*_{j}

*b*_{i} ⋅ *b*_{j}

R^{k}

*a*_{1,...,a}_{n}

*b*_{1,...,b}_{n}

See main article: Square root of a matrix. A matrix

*M*

*B*

*B*

*B*^{*}=*B*

*M*=*B**B*

*B*

*M*

*B*=

| ||||

M |

*M*

| ||||

M |

*M*

The non-negative square root should not be confused with other decompositions

*M*=*B*^{*B}

| ||||

M |

*M*=*B**B*

If

*M**>**N**>*0

| ||||

M |

*>*

| ||||

N |

*>*0

A positive semidefinite matrix

*M*

*M*=*LL*^{*}

*L*

*M*=*B*^{*B}

*B*=*L*^{*}

*M*

*L*

*M*=*L**D**L*^{*}

*D*

*L*

Let

*M*

*n* x *n*

*M*

- The associated sesquilinear form is an inner product: The sesquilinear form defined by

*M*

*\langle* ⋅ *,* ⋅ *\rangle*

C^{n} x C^{n}

C^{n}

*\langle**x,**y**\rangle**:*=*y*^{*M}*x*

*x*

*y*

C^{n}

*y*^{*}

*y*

*M*

*x*

*y*

C^{n}

*\langle**z,**z**\rangle*

*z*

*M*

C^{n}

- Its leading principal minors are all positive: The
*k*th leading principal minor of a matrix

*M*

*k* x *k*

*k*

*k*

A positive semidefinite matrix is positive definite if and only if it is invertible.^{[6]} A matrix

*M*

-*M*

See main article: Definite quadratic form. The (purely) quadratic form associated with a real

*n* x *n*

*M*

*Q**:*R^{n}*\to*R

*Q(x)*=*x*^{sf{T}Mx}

*x*

*M*

*\tfrac{*1*}{*2*}**\left(M*+*M*^{sf{T}\right)}

A symmetric matrix

*M*

More generally, any quadratic function from

R^{n}

R

*x*^{sf{T}Mx}+*x*^{sf{T}b}+*c*

*M*

*n* x *n*

*b*

*n*

*c*

*M*

A symmetric matrix and another symmetric and positive definite matrix can be simultaneously diagonalized, although not necessarily via 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

*M*

*N*

*\left(M*-λ*N\right)x*=0

*x*

x^{sf{T}Nx}=1

*N*

*Q*^{sf{T}Q}

*Q*

x=*Q*^{sf{T}y}

*Q\left(M*-λ*N\right)Q*^{sf{T}y}=0

*\left(QMQ*^{sf{T}\right)y=}λy

y^{sf{T}y}=1

*MX*=*NX*Λ

*X*

Λ

*X*^{sf{T}}

*X*^{sf{T}MX}=Λ

*X*^{sf{T}NX}=*I*

y^{sf{T}y}=1

*M*

*N*

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.

For arbitrary square matrices

*M*

*N*

*M**\ge**N*

*M*-*N**\ge*0

*M*-*N*

*M**>**N*

Every positive definite matrix is invertible and its inverse is also positive definite.^{[8]} If

*M**\geq**N**>*0

*N*^{-1}*\geq**M*^{-1}*>*0

*M*

*N*

If

*M*

*r**>*0

*r**M*

- If

*M*

*N*

*M*+*N*

- If

*M*

*N*

*M*+*N*

- If

*M*

*N*

*M*+*N*

- If

*M*

*N*

*MNM*

*NMN*

*MN*=*NM*

*MN*

- If

*M*

*A*^{*}*MA*

*A*

*M*

*A*

*A*^{*}*M**A*

The diagonal entries

*m*_{ii}

*\operatorname{tr}(M)\ge*0

*\left|m*_{ij}*\right|**\leq**\sqrt{m*_{ii}*m*_{jj}

and thus, when

*n**\ge*1

max_{i,j}*\left|m*_{ij}*\right|**\leq*max_{im}_{ii}

An

*n* x *n*

*M*

*\operatorname{tr}(M)**>*0 and

(\operatorname{tr | |

(M)) |

^{2}{\operatorname{tr}(M}^{2)}}*>**n*-1*.*

Another important result is that for any

*M*

*N*

*\operatorname{tr}(MN)**\ge*0

If

*M,**N**\geq*0

*MN*

*M**\circ**N**\geq*0

Regarding the Hadamard product of two positive semidefinite matrices

*M*=*(m*_{ij}*)**\geq*0

*N**\geq*0

- Oppenheim's inequality:

*\det(M**\circ**N)**\geq**\det**(N)**\prod\nolimits*_{i}*m*_{ii}*.*

*\det(M**\circ**N)**\geq**\det(M)**\det(N)*

If

*M,**N**\geq*0

*MN*

*M* ⊗ *N**\geq*0

If

*M,**N**\geq*0

*MN*

*M**:**N**\geq*0

The set of positive semidefinite symmetric matrices is convex. That is, if

*M*

*N*

*\alpha*

*\alpha**M*+*\left(*1-*\alpha\right)**N*

x

x^{sf{T}\left(\alpha}*M*+*\left(*1-*\alpha\right)N\right)x*=*\alpha*x^{sf{T}Mx}+*(*1-*\alpha)*x^{sf{T}Nx}*\geq*0*.*

This property guarantees that semidefinite programming problems converge to a globally optimal solution.

The positive-definiteness of a matrix

*A*

*\theta*

x

*Ax*

-*\pi**/*2*<**\theta**<*+*\pi**/*2

*\cos\theta*=

x^{T}Ax | = | |

\lVertx\rVert\lVertAx\rVert |

\langlex,Ax\rangle | |

\lVertx\rVert\lVertAx\rVert |

*,**\theta*=*\theta(x,Ax)*=*\widehat{x,Ax*

- If

*M*

*m*_{ij}

*m*_{ij}=*h(|i*-*j|)*

*M*

- Let

*M**>*0

*N*

*MN*+*NM**\ge*0

*MN*+*NM**>*0

*N**\ge*0

*N**>*0

- If

*M**>*0

*\delta**>*0

*M>\delta**I*

*I*

- If

*M*_{k}

*k* x *k*

*\det\left(M*_{k\right)/\det\left(M}_{k-1}*\right)*

- A matrix is negative definite if its
*k-*th order leading principal minor is negative when

*k*

*k*

A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative. It is however not enough to consider the leading principal minors only, as is checked on the diagonal matrix with entries 0 and −1.

A positive

2*n* x 2*n*

*M*=*\begin{bmatrix}**A**&**B**\* *C**&**D**\end{bmatrix}*

where each block is

*n* x *n*

*A*

*D*

*C*=*B*^{*}

We have that

z^{*}*Mz**\ge*0

z

z=*[v,*0*]*^{sf{T}}

*\begin{bmatrix}*v^{*}*&*0*\end{bmatrix}**\begin{bmatrix}**A**&**B**\* *B*^{*}*&**D**\end{bmatrix}**\begin{bmatrix}*v*\* 0*\end{bmatrix}*=v^{*}*Av**\ge*0*.*

A similar argument can be applied to

*D*

*A*

*D*

*M*

Converse results can be proved with stronger conditions on the blocks, for instance using the Schur complement.

*f(x)*

*n*

*x*_{1,}*\ldots,**x*_{n}

x^{sf{T}M}x

x

*M*

*f*

x

x

More generally, a twice-differentiable real function

*f*

*n*

*x*_{1,}*\ldots,**x*_{n}

In statistics, the covariance matrix of a multivariate probability distribution is always positive semi-definite; and it is positive definite unless one variable is an exact linear function of the others. Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution.

The definition of positive definite can be generalized by designating any complex matrix

*M*

*\Re\left(z*^{*}*Mz\right)**>*0

z

*\Re(c)*

*c*

x

*M*

x^{sf{T}M}x*>*0

x

x^{sf{T}M}x=*\sum*_{ij}*x*_{i}*M*_{ij}*x*_{j}

Consequently, a non-symmetric real matrix with only positive eigenvalues does not need to be positive definite. For example, the matrix

*M*=*\left[\begin{smallmatrix}*4*&*9*\* 1*&*4*\end{smallmatrix}\right]*

x^{sf{T}Mx}

x=*\left[\begin{smallmatrix}*-1*\* 1*\end{smallmatrix}\right]*

In summary, the distinguishing feature between the real and complex case is that, a bounded positive operator on a complex Hilbert space is necessarily Hermitian, or self adjoint. The general claim can be argued using the polarization identity. That is no longer true in the real case.

Fourier's law of heat conduction, giving heat flux

q

g=*\nabla**T*

q=-*Kg*

*K*

g

q

g

q^{sf{T}g<}0

g^{sf{T}Kg>}0

- Covariance matrix
- M-matrix
- Positive-definite function
- Positive-definite kernel
- Schur complement
- Sylvester's criterion
- Numerical range

- Book: Horn . Roger A. . Johnson . Charles R. . Matrix Analysis . . 978-0-521-54823-6 . 2013 . 2nd.
- Book: Bhatia, Rajendra . Rajendra Bhatia . Positive definite matrices . Princeton Series in Applied Mathematics . 2007 . 978-0-691-12918-1 .
- Bernstein . B. . Toupin . R. A. . 1962 . Some Properties of the Hessian Matrix of a Strictly Convex Function . . 210 . 67–72 . 10.1515/crll.1962.210.65 .

- 10.1002/9780470173862.app3 . Appendix C: Positive Semidefinite and Positive Definite Matrices . Parameter Estimation for Scientists and Engineers . 259–263. free .
- , p. 440, Theorem 7.2.7
- , p. 441, Theorem 7.2.10
- , p. 452, Theorem 7.3.11
- , p. 439, Theorem 7.2.6 with
*k*=2 - , p. 431, Corollary 7.1.7
- , p. 485, Theorem 7.6.1
- , p. 438, Theorem 7.2.1
- , p. 495, Corollary 7.7.4(a)
- , p. 430, Observation 7.1.3
- , p. 431, Observation 7.1.8
- , p. 430
- Bounds for Eigenvalues using Traces . Wolkowicz . Henry . Styan . George P.H. . Linear Algebra and its Applications . 29 . Elsevier . 1980 . 471–506 .
- , p. 479, Theorem 7.5.3
- , p. 509, Theorem 7.8.16
- Styan . G. P. . 1973 . Hadamard products and multivariate statistical analysis . . 6 . 217–240 ., Corollary 3.6, p. 227
- Book: Bhatia, Rajendra . Positive Definite Matrices . Princeton University Press . 2007 . 978-0-691-12918-1 . Princeton, New Jersey . 8 .
- Weisstein, Eric W.
*Positive Definite Matrix.*From*MathWorld--A Wolfram Web Resource*. Accessed on 2012-07-26