All Questions
Tagged with matrix-decomposition positive-semidefinite
53
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Decomposition of a positive semidefinite matrix in the form $Q = BB^H = CC^H$
Let $Q \in \mathbb{C}^{m \times m}$ be a given positive (semi)-definite matrix such that
$Q = CC^H,$
for some known matrix $C \in \mathbb{C}^{m \times r}$ having full column rank and orthogonal ...
- 15
0
votes
1
answer
37
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Positive semidefinite inequality: $(AXA^T)^+ \geq (AXA^T + Y)^{-1}$ on $\textrm{Im}(A)$
I asked a question earlier but it wasn't quite correctly stated, so I'll reset.
Let $A$ be an $n\times n$ matrix of rank $k<n$ and let $X,Y$ be two symmetric positive definite matrices. Let $Z^+$ ...
- 5,298
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1
answer
47
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Square root of non-symmetric positive semi-definite matrix.
Let $A$ be a $d \times d$ square positive semi-definite, not necessarily symmetric, real matrix in the sense that $v^TAv\ge 0$ for any vector $v\in\mathbb R^d$. How to show that there exists $B$ such ...
- 1,235
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83
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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
1
vote
1
answer
35
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If the symmetric part of a matrix is PSD then the matrix is PSD [duplicate]
My professor said this during my linear algebra class. But I am not sure how to prove this.
If A is any matrix, then it can be decomposed into a symmetric and anti-symmetric part like below-
$$
A = \...
- 81
1
vote
2
answers
423
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Decomposition of hermitian matrix as difference of positive semidefinite matrices
In my reference, Box 11.2, Page 512, Chapter 11, Entropy and Information, Quantum Computation and Quantum Information by Nielsen and Chuang, proof of the Fannes' inequality contains
here $\rho$ and $\...
- 7,674
1
vote
1
answer
188
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Difference of two positive semi-definite matrix still positive semi-definite?
Suppose $X\in\mathbf{R}^{n\times p}$ is a real matrix with full column rank($n\geq p$), $B \in \mathbf{R}^{n\times n}$ is a positive definite matrix, i.e all eigenvalue of matrix $B$ is positive, I ...
- 312
1
vote
1
answer
57
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Decomposition of a $2 \times 2$ matrix into the difference of 2 positive semi-definite matrices.
For the matrix $$A = \begin{bmatrix}
0 & 1 \\
1 & 0 \\
\end{bmatrix}$$ with norm $||A|| = \sum_{i,j = 1,2} |a_{ij}|$
Show any decomposition of $A = C - B$ with $B, C$ being ...
- 39
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votes
1
answer
96
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Finding the positive projection in the frobenius norm
I am trying to find the positive projection in the frobenius norm of a real matrix. Consider the following matrix $\hat{Z}$:
\begin{bmatrix}1&2&0\\0&5&0\\0&8&9\end{bmatrix}
I ...
1
vote
0
answers
49
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Inequality for PSD block matrices
Given a positive definite matrix $M\in\mathbb{R}^{(2n)\times(2n)}$ of the following block form
$$M = \begin{bmatrix}A & X\\X^\top& B\end{bmatrix},$$
where $A, B\in\mathbb{R}^n$ are positive ...
- 73
0
votes
1
answer
99
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Is the matrix for Cholesky decomposition semidefinite or definite?
So we have a matrix $A$. We need to check if it is positive definite (numerically). At lectures we have done that with brute-force Cholesky decomposition, and if any of the square roots is not defined,...
- 2,527
0
votes
1
answer
476
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Cholesky decomposition for symmetric positive semi-definite matrices
On page 5 here: https://stanford.edu/class/ee363/lectures/lmi-s-proc.pdf
$A$ and $B$ are decomposed into $A^{1/2} A^{1/2}$ and same for $B$.
Is this from Cholesky decomposition? Can someone prove ...
- 729
1
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2
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55
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Conditions for positivity of this symmetric real matrix
I have the following real symmetric matrix $M$ of size 3:
\begin{align}
M =
\begin{pmatrix}
a & b & c \\
b & d & e \\
c & e & f
\end{pmatrix},
\end{align}
for real parameters ...
- 403
0
votes
1
answer
98
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On what conditions for ${\bf C}$, such that $trace({\bf AC}) \geq trace({\bf BC})$, given that $trace({\bf A}) \geq trace({\bf B})$?
Edit:
Given two real symmetric matrices ${\bf A}$ and ${\bf B}$, of size $n \times n$, such that their traces satisfies $trace({\bf A}) \geq trace({\bf B})$. I wish to find a general $n \times n$ ...
- 41
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votes
1
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55
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If I know the ODE of a PSD matrix, how can I find the ODE of its square root matrix?
I know that the ODE that describes how the covariance matrix $P$ changes over time is
$$ \dot{P} = A P + P A^T $$
where both $A$ and $P$ are time-varying $n \times n$ matrices.
Since $P$ is a ...
