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Results tagged with linear-algebra
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user 38620
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
2
votes
How do we show this matrix has full rank?
I use Christian Remling idea,In fact,I can find the matrix $$B_{ij}=\min{\{i,j\}}$$eigenvalue is
$$\dfrac{1}{4\sin^2{\dfrac{j\pi}{2(n+1)}}},j=1,2,\cdots,n$$
proof:
then we have
$$B=\begin{bmatrix}
…
- 4,280
3
votes
1
answer
728
views
A similar Cauchy-Schwarz inequality with linear-algebra
Let $A$ be matrix in $M_{n}$ (i.e., $n\times n$ complex matrices), and $\|A\|\le 1$, we call it a contraction.
Assume that $A$ and $B$ are contractions such that
$I-AA^*$ and $I-BB^*$ are positive-d …
- 4,280
13
votes
3
answers
3k
views
How do we show this matrix has full rank?
I met with the following difficulty reading the paper Li, Rong Xiu "The properties of a matrix order column" (1988):
Define the matrix $A=(a_{jk})_{n\times n}$, where
$$a_{jk}=\begin{cases} …
- 4,280
5
votes
0
answers
2k
views
A stronger Cauchy-Schwarz inequality for traces of compression matrices
Assume that $A$ and $B$ are contractions, so
$I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let
$C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:
$$Tr\left(\frac{1}{1-AA^T}\right) …
- 4,280