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Results tagged with quadratic-forms
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user 8430
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
2
votes
if Y-X is positive semi-definite, are the eigenvalues of Y bigger?
At this point I find it worth mentioning the following facts. Let $X$ and $Y$ be arbitrary square complex matrices. Let $|X|=(X^*X)^{1/2}$ be the matrix absolute value. Then, we may have the following …
- 28.3k
7
votes
Accepted
Maximizing quadratic form on the hypercube
This maxQP problem is hard, it includes MaxCUT as a special case---see this paper by M. Charikar on MAXQP. Having $Q$ be positive definite does not really help (take the MAXQP problem in Charikar's pa …
- 28.3k
3
votes
Accepted
Block Covariance Matrix - Positive Definite? (Quadratic Optimization)
If $C$ is positive semidefinite, then so is $\begin{bmatrix} C & C\\ C & C\end{bmatrix}$ for the simple reason that it is nothing but the Kronecker product of $\begin{bmatrix} 1 & 1\\ 1 & 1\end{bmatri …
- 28.3k
11
votes
Accepted
A spectral inequality for positive-definite matrices
$\newcommand{\trace}{\operatorname{trace}}$
The result below mentions a reasonably improved inequality.
Let $m = \frac{\trace(A)}{n}$, and $s^2= \frac{\trace(A^2)}{n}-m^2$. Then, Wolkowicz and Styan …
- 28.3k
2
votes
Optimization problem on trace of rotated positive definite matrices
To expand on my comment (and given the update by the OP), it is clear that $R=UP^T$ (where $A=PDP^T$ and $B=ULU^T$) maximizes the trace. This follows because
$\text{tr}(RAR^TB) \le \langle\lambda^\dow …
- 28.3k
14
votes
Accepted
On the positive definiteness of a linear combination of matrices
The following recent paper: "An exact duality theory for semidefinite programming based on sums of squares" by I. Klep, and M. Schweighofer (both are on MO I think) addresses exactly your question: Wh …
- 28.3k