Questions tagged [positive-definite]
For questions about positive definite real or complex matrices. For questions about positive semi-definite matrices, use the (positive-semidefinite) tag.
1,700
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Relating two seemingly different definitions of Riemannian metric.
I've been studying Riemannian geometry from John M. Lee's book on the same. My aim is to understand the geometry of Hermitian positive semidefinite matrices when viewed as a Riemannian manifold.
In ...
- 27
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1
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23
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Existence of a linear map to the space of SPD matrices
Does there exist a linear map from the set of 24x29 Matrices to the set of Symmetric Positive definite (SPD) 24x24 Matrices? I understand that matrix multiplication is a linear map but from the matrix ...
- 35
4
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1
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Prove that $\text{tr}(B_1^{-1} B_2) \geq \text{tr}((A^\text{T} B_1 A)^{+} A^\text{T} B_2 A)$
Suppose that we have two real and positive definite $n \times n$ matrices $B_1$ and $B_2$ and that $A$ is an arbitrary real $n \times n$ matrix. Running some numerical tests by generating random ...
- 43
1
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28
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Nature of a certain invariant on smooth field of positive definite matrices
I apologize if this question is too open for this forum.
Denote $g$ a positive definite matrix field (defined over, say, $\mathbb R^n$), and $g_{ij}$ its components in some basis (the same for all $g(...
- 121
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3
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72
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Matrix with entries $A_{ij} = u^{|i-j|} - u^{i+j}$ is positive semidefinite when $u \in (0, 1)$?
Consider the square $n \times n$ matrix $A_n$ with entries $u^{|i-j|} - u^{i+j}$ where $u \in (0, 1)$.
Is it true that $A$ is positive semidefinite?
In the case $n = 1$, we have $A_1 = 1-u^2 \geq 0$.
...
- 3,102
2
votes
2
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69
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Showing matrix C has spectral radius less than 1
We are given that $A$ is symmetric and positive definite, and $B$ is such that $A-B-B^T$ is also symmetric and positive definite. We are asked to show that $C=-(A-B)^{-1} B$ has spectral radius less ...
- 33
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39
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Positive definite matrix and inverse
The book asserts that $\Psi$ is a positive definite matrix and expresses it as $\Psi^{-1} = P'P$, where $P$ is a square non-singular matrix. The implication is that, due to the positive definiteness ...
- 133
2
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1
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68
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Positive/Negative Definite Functions Confusion
I have the Lyapunov function $V(x_1,x_2) = x_1^2+x_2^2$ for the nonlinear system:
$$\dot{x_1} = x_2$$
$$\dot{x_2} = -x_1 - x_2-x_2^3$$
Now obviously $V$ is a positive definite function but in order to ...
- 475
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64
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Is this matrix always semi-definite?
Assume I have a 3D vector $v$ and $\times$ is a dyadic product.
Is matrix
$$
M = v \times v'
$$
always positive semi-definite?
Computing $x^T M x$ gives us $(x_1 v_1 + x_2 v_2 + x_3 v_3)^2$, which is ...
1
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36
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Quadratic Form and its Matrix, Associated bilinear form for positive definite quadratic form is nondegenerate.
I have a question regarding the quadratic forms and their associated matrices. For some reason, google keeps telling me that this matrix is symmetric. The definition I am using is that a quadratic ...
- 391
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0
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49
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Show that the solution to $Ax = b$ is a critical points of $f(x)$
Given the system of linear equations (the matrix of the system is symmetric and positive definite):
$$\left\{\begin{matrix}
10x + 5y = 5\\
5x + 3y = 4
\end{matrix}\right.$$
The solution to the system ...
0
votes
1
answer
32
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generalized matrix inequality for complex Hermitian matrices
Assume having a symmetric real matrix $A$ and a skew-symmetric matrix $\Delta = [0 1; -1, 0 ]$, such that the following generalized matrix inequality holds in the PSD sense:
$$\pm \frac{i}{2} \Delta\...
- 55
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43
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Calculate the derivative of the Frobenius norm of matrix logarithm
Let $h(X)=\|\log(X)\|_{F}^2$ where $X\in\mathbb{S}_{++}^d$ is a $d$ by $d$ symmetric positive definite matrix, $\log$ is the matrix logarithm and $F$ stands for the common Frobenius norm. In section 6....
- 3
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15
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Ordering positive definite matrices with diagonal matrices
Let $A \succ 0$ be a Hermitian positive definite matrix. I'm trying to understand the claim that there exists a (diagonal) positive definite matrix $B \succ 0$ such that $A \prec B$, that is, $B-A$ is ...
- 57
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31
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Finding the closest positive definite matrix
Let $L$ be the symbolic $n \times n$ lower triangular matrix and $S$ a real symmetric $n \times n$ positive semidefinite matrix. Fix a real symmetric $n \times n$ positive definite matrix $A$ with the ...
- 187
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If $A^2\succeq B^2$ then $A\succeq B$ [duplicate]
Let $A$ and $B$ be $n×n$ symmetric matrices such that $A,B\succeq0$ ($A$ and $B$ are positive semidefinite) and $A^2\succeq B^2$. Is the following inequality true? $$A\succeq B$$ In this answer, it's ...
- 4,569
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27
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Necessary and sufficient condition for Lyapunov stability
Suppose $J $ is a real square matrix and matrix $V = V^\textsf{T}$ is positive-definite. Define $ A = V J $.
Can we show that $J ^\textsf{T} + J$ is negative-definite if, and only if, $A^\textsf{T} P +...
1
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30
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Is $\lambda_{\min}(A) \geq \lambda_{\min}(B)$ when $A > B$ where $A$ and $B$ are positive semidefinite matrices?
Is $\lambda_{\min}(A) \geq \lambda_{\min}(B)$ when $A > B$ where $A$ and $B$ are positive semidefinite matrices? Here $A > B$ means $A-B$ is positive definite matrix.
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If $A=M^{-1}A^tM$ for some $M>0$, then there exists an invertible matrix $P$ such that $P^{t}MP=I$ and $P^{-1}AP$ is diagonal.
Problem
Let $V=\mathbb R^n$ be the space of column vectors, and $M$ a positive definite symmetric $n\times n$ real matrix. Suppose the matrix $A\in M_n(\mathbb R)$ satisfies $MAM^{-1}=A^t$. Show that ...
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Does $0\preceq A \preceq B$ imply $f(A) \preceq f(B)$ for concave(?) functions?
It seems to be a fairly common exercise to show that
$$
0\preceq A \preceq B \implies \sqrt A \preceq \sqrt B
$$
for positive semi-definite matrices. On the other hand, in general,
$$
0\preceq A \...
- 6,964
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0
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34
views
Prime numbers and a positive definite matrix?
Probably this has nothing to do with prime numbers, I just experimented a little bit with it and wanted to share it, in case someone has an idea.
Let
$$p_n := n\text{-th prime number , }[a,b]:= \frac{...
- 609
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1
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28
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Proving that a matrix of Gaussian distances is positive definite
I have a square matrix with elements $B_{ij} = e^{-c(i-j)^2}$ for some $c > 0$, i.e. a sort of matrix of Gaussian distances. How would I go about proving that this matrix is positive definite?
I ...
- 485
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4
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SVD of "normalized" SPD matrix: $P=\sigma \rho \sigma^T$. Relate SVD of $rho$ to the SVD of $P$?
$P\in\mathbb{R}^{n \times n}$ is a symmetric positive definite (SPD) covariance matrix, and can be factorized into standard deviations and correlations as $P=\sigma \rho \sigma^T=\sigma \rho \sigma$. ...
- 21
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10
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SVD or Cholesky on sum of SPD matrices
Let $A$ and $B$ be symmetric positive definite (SPD) matrices and $C=A+B$. I know the SVD or Cholesky decomposition of A and B, $A=U_A\Sigma_AU_A^T=L_AL_A^T$ and $B=U_B\Sigma_BU_B^T=L_BL_B^T$.
Can I ...
- 21
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22
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Gradient of quadratic form with positive definite matrix in terms of Jacobian
I have recently encountered the following in a paper: Given $(\boldsymbol{z}-\boldsymbol{y(x)})^TU(\boldsymbol{z}-\boldsymbol{y(x)})$ where $U$ is a positive definite matrix independent of $x$ and $\...
- 75
1
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104
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Lower bound for matrix vector product
Let $\bf{A}$ be a semi-positive matrix, $\bf{I}$ is the identity matrix, and $\alpha \geq 0$. Let $\bf{x}$ and $\bf{y}$ be two vectors. I'm looking for a lower bound for the following quantity:
$${\bf ...
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Conditions that lead empirical set of covariance matrices to follow optimal transport path?
I have a statistics/learning problem where I compute the covariance matrix of a random variable $Y \in \mathbb{R}^n$ conditional on different values of a second variable $X \in \mathbb{R}$. We refer ...
- 202
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1
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52
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Is the following Hankel matrix positive definite?
The matrix whose positive definiteness I am investigating is the following one
$$
H_k(p)=\begin{bmatrix}1 & \frac{1}{p+1} & \frac{1}{p+2} & \frac{1}{p+3} & ... & \frac{1}{p+k-1} \...
- 9
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When is the product of a skew-symmetric matrix $\mathbf{J}$ and a symmetric postive definite matrix $\mathbf{A}$ symmetric positive definite?
Let $\mathbf{J}$ be a skew-symmetric matrix and $\mathbf{A}$ be a symmetric positive definite matrix. How can we construct $\mathbf{J}$ and $\mathbf{A}$ such that $\mathbf{J}\mathbf{A}$ is symmetric ...
- 129
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votes
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40
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Can $\operatorname{Tr}[A^{-1} BAB^\top]$ be shown to be always positive if $A$ is real and positive definite? [closed]
Let $A$ be a real symmetric positive definite matrix and $B$ is a real matrix with all eigenvalues zero. Can we prove or disprove that $\operatorname{Tr}[A^{-1} BAB^\top]$ is a positive number?
- 261
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Equivalence of Matrix Determinant Maximization Problems?
I am interested in an optimization problem involving parameters $\boldsymbol{\mu}$ and $\boldsymbol{\pi}$, both strictly positive stochastic vectors, and variable $n \times n$ matrix $W$:
\begin{...
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1
answer
33
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Positivity of an $n \times n$ matrix in $M_n(B(\mathcal H))$
Let $\mathcal H$ be a Hilbert space and $B(\mathcal H)$ be the set of all bounded operators on $\mathcal H$. Let $B \subset B(\mathcal H)$ be a $C^*$-subalgebra of $B(\mathcal H)$. Let $\xi \in H$ be ...
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If $A,B\in GL_n(\mathbb{C})$ are positive matrices. Prove that the following holds
If $A,B\in GL_n(\mathbb{C})$ are positive matrices. Prove that, there is $\lambda\in\sigma(A^{-1}B)$ such that the following holds $$\langle Ax,x\rangle\ge\lambda^{-1}\langle Bx,x\rangle\ \forall x\in\...
- 1,074
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Diagonal submatrices of the inverse of a $p \times p$ block matrix
Let $X$ be a square, symmetric, positive definite matrix that can be decomposed into $p\times p$ block matrices:
$$X = \begin{bmatrix} X_{11} & X_{12} & \ldots & X_{1p}\\
X_{21} & X_{...
- 1,946
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Can we say whether a matrix is positive definite from the matrix propeties [duplicate]
By definition, a symmetric matrix $\mathrm{A} \in \mathbb{R}^{n \times n}$ is positive definite if for all $\mathrm{x} \in \mathbb{R}^n \setminus \{\mathrm{0}\}$, $\mathrm{x}^{\mathrm{T}} \mathrm{A} \...
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Upper bound on the trace of the product of real PSD matrices involving the trace of one of them
I am looking at the negative log-likelihood of a multivariate Gaussian $X \sim \mathcal{N}(\mu, \Sigma)$:
$$const + (n/2)\log|\Sigma| + (1/2)Tr(\Sigma^{-1}S),$$
where $S = (1/n)\sum(X_i-\mu)(X_i-\mu)'$...
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3
votes
0
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90
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Intuitive difference between optimal transport distance and Fisher information distance
Let me start by saying I'm not a mathematician but a biologist with an interest in mathematics.
I have a set of covariance matrices and I am interested in studying their geometry in the Symmetric ...
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0
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123
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How to calculate the eigenvalues of matrix with entries that are continuous functions of matrix?
I met the following problem during my research, could anyone please give me some help?
Let $A(x) \in \mathbb{R}^{2 \times 2}, x\in \mathbb{R}$ be a symmetric matrix in the form
\begin{align}
A(x) = \...
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1
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Does the product of a PSD matrix and a positive eigenvalue matrix have positive eigenvalues?
It's known that if $A$ is a positive definite (symmetric) matrix and $B$ is positive semi-definite (not necessarily symmetric), then $AB$ has non-negative eigenvalues. It follows from knowing that $A$ ...
- 665
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Distance between two positive semidefinite matrices
Given two positive semidefinite matrices X, and Y. My question is, can we use von Neumann or the log-Determinant divergences to measure the distance between the two matrices. Most Manifold measure ...
- 562
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1
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A inequality satisfied over an ellipsoid
Let $A$ be a positive definite matrix and $v \in \mathbb{R}^n$ be a given vector. Suppose there exists a $y \in \mathbb{R}^n$ such that $0 < y^{\top}Ay \leq 1 $ and the following holds:
$$
\langle ...
- 113
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How to prove this inequality containing cofactors of a matrix?
I want to prove that the matrix $A$ of order $n$ with $(a_{ij})=\frac{2^{i+j}-2^{|i-j|}}{3}$ admits a Cholesky factorization. $A$ is trivially symmetric, but when trying to show it is positive ...
0
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1
answer
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Prove: If matrix $A\in\mathbb{C}^{m\times n}$ with $m>n$ is full rank and $B\in\mathbb{C}^{n\times n}$ is diagonal, $ABA^H$ is positive semi-definite
If matrix $A\in\mathbb{C}^{m\times n}$ with $m>n$ is full rank and $B\in\mathbb{C}^{n\times n}$ is diagonal with $det(B)\ne0$, and $(B)_{ii}>0$ how can we prove that $ABA^H$ is positive semi-...
- 15
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vote
1
answer
75
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determinant of symmetric block matrix with positive definite diagonal blocks
I have a matrix of the form
$B = \left[\begin{array}{cccc}
A_1 & C^T\\
C & A_2\\
\end{array} \right]$
Where $A_1,A_2$ and $C$ are all square, and $A_1,A_2$ are symmetric positive definite. ...
- 597
0
votes
1
answer
44
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Analog of Sherman–Morrison matrix inversion formula for $(A-bx^T)^T (A-bx^T)$
Let $A$ be a real $n \times d$ matrix, $\vec{b} \in \mathbb{R}^n$, and $\vec{x} \in \mathbb{R}^d$. I'd like to find a simple formula for$$
F(\vec{x}) = \left(\left(A-\vec{b}\vec{x}^T\right)^T \left(A-...
- 2,233
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0
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74
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Difference of inverese matrices
Let $A$, $B$ be symmetric positive definite matrices. We know
$$
A^{-1}-B^{-1}=-C
$$
for positive semidefinite matrix $C$.
I want to show that $A-B$ is positive semidefinite. Here's my take:
\begin{...
- 13
0
votes
1
answer
90
views
Proving that a symmetric real matrix with a specific structure is positive definite.
Let $H$ be an $N\times N$ symmetric matrix with the following structure:
\begin{equation}
[H]_{n,m}\triangleq\begin{cases}
{\left|x_{n}\right|}, &\text{if}\ n=m,\\
-\mathsf{Re}{\left(...
- 33
1
vote
1
answer
26
views
Any symmetric normed ideal $\mathfrak{a}$ on $\mathcal{H}$ is linearly generated by its positive elements
I have some questions about the proof of this statement in the book "Elements of Noncommutative Geometry" by Garcia-Bondía.
A ideal $\mathfrak{a}\subset K(\mathcal{H})$ is called ...
0
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0
answers
13
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Checking whether matrix is PD vs computing PD completion
Let $E$ be a subset of entries of a real symmetric $n \times n$ matrix. We want to find a positive-definite matrix $X$ such that $X_{i,j}=M_{i,j}$ for all $(i,j) \in E$ for a fixed positive ...
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0
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25
views
Given $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times p}$ with $rank(B)=p$ and $C(B)\subset C(A)$. Prove $B'AB$ is positive definite?
I want to show:
Given $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times p}$ with $rank(B)=p$ and $C(B)\subset C(A)$. Prove $B'AB$ is positive definite, where $C(B)$ denotes the column ...
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