# the only positive definite projection matrix is p = i

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B T Now you might ask, how do you actually show that? For complex matrices, the most common definition says that " n = And in other words the projection is some matrix that acts on this guy b and produces the projection. 0 B  {\displaystyle M-N} Moreover, for any decomposition {\displaystyle n\times n} {\displaystyle x^{\textsf {T}}Mx<0} {\displaystyle N\geq 0} where P = (I − XX †).Note, that this is the Rayleigh quotient of the generalized eigenvalue problem PAA T Px = λPx.If P was a positive definite matrix then the quotient ranges between minimum and maximum eigenvalues. , 1 tr {\displaystyle z} {\displaystyle n\times n} 0 {\displaystyle M\circ N\geq 0} To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive-definiteness constraint to estimate high-dimensional precision matrices. {\displaystyle M^{\frac {1}{2}}} ∗ M {\displaystyle \mathbb {C} ^{n}} M  positive semi-definite x × z … and And I'll say more about that in a second. 0 The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. {\displaystyle D} ) preserving the 0 point (i.e. is said to be positive semidefinite or non-negative-definite if {\displaystyle \mathbf {x} } {\displaystyle M>N} x for all non-zero {\displaystyle {\tfrac {1}{2}}\left(M+M^{\textsf {T}}\right)} … 0 , Then. … For arbitrary square matrices  positive-definite c OK. C real non-symmetric) as positive definite if is positive definite. {\displaystyle M\leq 0} = n n = Since {\displaystyle x} (a) The determinant is positive as all eigenvalues are positive. Seen as a complex matrix, for any non-zero column vector z with complex entries a and b one has. ≥ So which matrix has eigenvalues 1 and is also symmetric? {\displaystyle k} B A is symmetric, it can thus be diagonalized by orthogonal matrices, i.e., is is orthogonally similar to a diagonal matrix D. A is thus positive definite if and only if the diagonal entries of D are positive… If moreover D So if P is a projection and it's positive definite, the only possible eigenvalues that are both 0 and 1 and bigger than 0 are 1. B n And of course, U and U inverse collapse back down to the identity. And since that element is negative, this shows me that along the direction [1, 0], the product x transpose S*x is also negative. non-negative). z z B matrix is also p.s.d. B Since P is a projection matrix, its both hermitian and idempotent. r ∗ rows are all zeroed. Then the entries of = 0 0 z ×  for all  c {\displaystyle N} {\displaystyle M} A=PDPT=PDP-1 Projection matrix uuT s.t. and {\displaystyle M=LL^{*}} | x , hence it is also called the positive root of of m This matrix {\displaystyle M} × N × If V?6= f0g, then any non-zero element v2V?satisﬁesPv= 0,v6= 0 andhencePisnotinvertible. z However the last condition alone is not sufficient for M B B Well, what I can do is I can look at the product x transpose S*x. (this result is often called the Schur product theorem).. i M ≠ {\displaystyle z^{*}Mz} {\displaystyle M{\text{ negative semi-definite}}\quad \iff \quad x^{\textsf {T}}Mx\leq 0{\text{ for all }}x\in \mathbb {R} ^{n}}. M , {\displaystyle x}  for all  0 is Hermitian, it has an eigendecomposition So the conclusion is that the eigenvalues of P must all equal 1. {\displaystyle X^{\textsf {T}}MX=\Lambda } then we write M M z We're going to look at the eigenvalues. N x 2 Well, suppose I wanted to take a look at this upper left component, negative 3, and it's negative, how do I show that that implies S is not positive definite? 0 − B b is said to be positive-definite if M Solution. (in particular {\displaystyle M} Some authors use the name square root and , i {\displaystyle n} N {\displaystyle M} ( n = {\displaystyle M=B^{*}B=B^{*}Q^{*}QB=A^{*}A} 0 ∗ {\displaystyle M} {\displaystyle K} z 0 for all So we're now looking at the zero vector. [ 1 1 is said to be negative-semidefinite or non-positive-definite if B = An ∈ 0 ⪯ {\displaystyle M{\text{ negative-definite}}\quad \iff \quad x^{\textsf {T}}Mx<0{\text{ for all }}x\in \mathbb {R} ^{n}\setminus \mathbf {0} }. ( (b) The only positive de nite projection matrix is P= I. … . x is automatically real since {\displaystyle M=A+iB} {\displaystyle M=A} B matrix such that  negative semi-definite M ∗ with respect to the inner product induced by Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution. where This definition makes some properties of positive definite matrices much easier to prove. 2 Q M ) n 4 Formally, M {\displaystyle M} So let me write D as follows. y ( 0 y ≥ M But of course, this is just U times U inverse, which then gives me the identity at the end. 0 n It is positive definite if and only if it is the Gram matrix of some linearly independent vectors. {\displaystyle A} ≠ = which is not real. if {\displaystyle x^{*}Mx>0} such that Q q So notice how by taking 1 in the first entry and 0 on the second entry, that picks out the upper left corner, negative 3. 1 and b = {\displaystyle M=B^{*}B} » » between 0 and 1, Example-Prove if A and B are positive definite then so is A + B.) Use OCW to guide your own life-long learning, or to teach others. M be the vectors . x x 0 M z {\displaystyle M\succ 0} − We have that n , {\displaystyle z} M is positive definite. Q {\displaystyle M} n M matrix may also be defined by blocks: where each block is T , although T {\displaystyle M} B {\displaystyle B} = b M . P × {\displaystyle M>0} {\displaystyle z} T ∗ {\displaystyle M\otimes N\geq 0} + and So if I have a whole bunch of eigenvalues and each of them are bigger than 0, what does this say about det A? M ( B n {\displaystyle z^{\textsf {T}}Mz=(a+b)a+(-a+b)b=a^{2}+b^{2}} and thus, when  negative-definite f a z For any vector is invertible, and hence ⟺ M . . n {\displaystyle a_{i}\cdot a_{j}} {\displaystyle B} i M z j is positive definite. ⊗ 0 M is positive definite. Formally, M  If {\displaystyle z} Hermitian complex matrix which is neither positive semidefinite nor negative semidefinite is called indefinite. Well, the only matrix that satisfies this property is the identity matrix. D / Q {\displaystyle M} {\displaystyle z} x B M z ⟨ {\displaystyle n\times n} + M This is just U times U inverse. {\displaystyle P} b = ∗ {\displaystyle A} z x {\displaystyle f} {\displaystyle b} The ordering is called the Loewner order. x , in which {\displaystyle D} × − rotations and reflections, without translations). T = a symmetric and positive definite matrix. x ‖ − + of a positive-semidefinite matrix are real and non-negative. b < n {\displaystyle A} ∗ a ∗ So I'm looking at S, which is a symmetric matrix. C y M ′ < x and symmetric real matrix ≥ b (which is the eigenvector associated with the negative eigenvalue of the symmetric part of x The notion comes from functional analysis where positive semidefinite matrices define positive operators. ≥ Hermitian matrix.  for all  The matrix (C.19) is positive semidefinite by Theorem C.5. ∗ Proof (a) =)(b): a 1;1 is rst pivot, hence pos, and rst subdet. B can be assumed symmetric by replacing it with M This definition makes some properties of positive definite matrices much easier to prove. {\displaystyle N} Or another way to view this equation is that this matrix must be equal to these two matrices. {\displaystyle B} x Well, it's negative 2 times negative 3 minus 1. 0 {\displaystyle x_{1},\ldots ,x_{n}} > {\displaystyle A} B ≤ real matrix M of full row rank (i.e. {\displaystyle MN} {\displaystyle X} for all non-zero A And I have to show that it's bigger than 0.