# Questions tagged [matrix-analysis]

For question about matrices and their algebraic properties. Together with [tag:linear-algebra] if necessary.

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### The necessary and sufficient constraint of $A$ such that $\text{tr}(A^n)\geq0$?

I'm looking for a sufficient and necessary constraint for the complex square matrix $A$ such that $\text{tr}(A^n)\geq0$ for $n=1,2,3...$ Up to now, it seems that the eigenvalues of $A$ must come in ...
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### Projection onto the Stiefel manifold and the orthogonal Procrustes problem

Let $m$ and $n$ be positive integers such that $m \ge n$, the case with $m >n$ being particularly interesting to us. The Stiefel manifold is \mathbb{S}^{m, n} = \{X \in \mathbb{R}^{...
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### Bhatia, Ex 7.1.12: Principal angles between subspaces

I am unable to solve this problem in Matrix Analysis by Rajendra Bhatia: I can't see how to prove that $\theta_i$ are decreasing. I also have no intuition for why the angles are defined this way. ...
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### Property regarding two positive definite matrices

Suppose $A$ and $B$ are two $n \times n$ symmetric positive definite real matrices and let $M$ be an $n \times m$ real matrix of full rank. Suppose that $A - B$ is positive definite. Then show that ...
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### If $T=\begin{bmatrix}x&Y^*\\Y &Z\end{bmatrix}$, $Z\ge 0$ $n\times n$ matrix, $x>0$ and $Y$ is $n\times 1$ vector, $T\ge 0$ iff $Y\in\text{ran }(Z)$

This is what I have to prove: If $T=\begin{bmatrix} x & Y^* \\ Y & Z\end{bmatrix}$ where $Z$ is positive semidefinite $n\times n$ matrix, $x>0$ and $Y$ is $n\times 1$ column vector then $T$...
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### Convexity of a special kind of set

Let $\mathcal{H}$ be a separable Hilbert space and $T\in B(\mathcal{H})$. Is the following set convex \begin{align} \{\langle Te_1,e_2\rangle:\{e_1,e_2\} \text{ are orthonormal set in } \mathcal{H}\}? ...
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### Convertor matrices

Let us consider $\mathbb{C}^n$, the set of all $n$-tuples of complex numbers. We know that $(\mathbb{C}^n,+,\cdot)$ is a unital commutative complex algebra. Suppose that $\diamond$ is an ...
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### Does $\|A^k\| \leq a k^{n-1} \rho^k(A)$ hold?

Given a non-singular matrix $A \in \mathbb{R}^{n\times n}$, let $\rho(A)$ be its spectral radius. Then, $\forall k \in \mathbb{N_0}$, can we find a constant $a > 0$ such that the following ...
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### What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])?

For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this link for detail), i.e., say $a$ is a complex ...
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### Proof that sum of k-eigenvalues is convex

I saw a post https://mathoverflow.net/questions/98367/a-sum-of-eigenvalues that said It is well-known that $\sum^{r}_{i = 1} \lambda_{i}(X)$ is convex. and I saw an explanation in Boyd and ...
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### Change of eigenvalues under near orthogonal matrix multiplication

Let $A,B\in\mathbb{R}^{d\times d}$ be positive definite diagonal matrices. Let $\Phi\in\mathbb{R}^{n\times d}$ satisfy $\text{rank}(\Phi) = n \leq d$ and be such that $\Phi\Phi^T = I_n$. When $d = n$, ...
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### If $AB^∗$ and $B^∗A$ are both normal, show that $B A^∗A = A A^∗B$ [closed]

This might be an easy question, but I haven't been able to prove it. Prove that if $A B^*$ and $B^* A$ are normal matrices, then $B A^* A = A A^* B$ Any help is appreciated. Here, $A^*$ is the ...
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### What's the singular value of a symmetric matrix plus identy matrix? $A+\lambda I$

Suppose, we know the singular values of a symmetric matrix $A$ as $\{\sigma_1,\cdots,\sigma_n\}$. What is the singular values of the matrix $A+\lambda I$?
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### Lower bound of the smallest singular value of product of two matrices

Suppose that $X, Y \in \mathbb{R}^{m \times n}$ both have full row rank ($m < n$), is it possible to obtain the lower bound of $\sigma_{\min} (XY^{T})$? The smallest non-zero singular value of AB ...
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