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Jun
25
revised How do I prove that the inverse of the matrix, M, below has all elements greater than or equal to 0?
added 10 characters in body
Jun
24
answered How do I prove that the inverse of the matrix, M, below has all elements greater than or equal to 0?
Jun
24
comment Point closest to a set four of lines in 3D
@elect Would you be more specific? And do you know the basic stuffs, such as how to multiply two matrices (in particular, how to calculate the outer product of two vectors) and how to find the norm of a vector?
Jun
15
comment irreducible, diagonally dominant matrix
@par You are reading between the lines, I guess? The OP says "irreducible, diagonally dominant matrix", not "irreducibly diagonally dominant matrix". He/she even cared to add a comma between "irreducible" and "diagonally dominant".
Jun
7
answered Order $n^2$ different reals, such that they form a $\mathbb{R^n}$ basis
Jun
3
comment Characterization of positive definite matrix with principal minors
@Chappers In the first paragraph, we have taken for granted the fact that every Hermitian matrix is unitarily diagonalisable. So, the determinant of a positive definite matrix, i.e. the product of the eigenvalues, is obviously positive.
Jun
3
comment Characterization of positive definite matrix with principal minors
@Chappers When $A$ is positive definite, every leading principal submatrix of $A$ is positive definite too.
Jun
3
answered Is it possible to have an $n\times n$ real matrix $A$ such that $A^TA$ has an eigenvalue of $-1$?
Jun
2
revised Determinant of block tridiagonal matrices
added 35 characters in body
Jun
2
comment If I have a diagonal matrix, is it necessarily the product of two other diagonal matrices?
Then you should edit what you intended into your question. In particular, in the question title and at the end of the second sentence, "two other diagonal matrices" should read "two non-diagonal square matrices".
May
31
answered How is $X(X^{\prime}X)^{-}X^{\prime}$ symmetric?
May
30
comment Do $J$, the all-ones matrix of even order, always have eigenvectors consisting of entries $-1, 1$ only?
Hmm, why are columns of a Hadamard matrix eigenvectors of $J$? By the way, the existence of Hadamard matrices of order $4k$ for every $k$ is still a (rather famous) conjecture.
May
30
comment Show that there do not exist nonsingular matrices $P,Q ∈ M_{n×n}(F )$ satisfying $PAQ = A^T$ for all $A ∈M_{n×n}(F )$.
Have you tried to see what the implications are when $A=E_{ij}$, the matrix with a 1 at the $(i,j)$-th entry and zero elsewhere? Exercises like this can usually be solved with this approach.
May
30
revised $A,B$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?
added 2 characters in body; edited title
May
30
comment $A,B$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?
The end note is wonderful. +1
May
29
revised Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$.What is numerical range $A$
added 22 characters in body
May
29
revised Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$.What is numerical range $A$
added 98 characters in body
May
29
comment Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$.What is numerical range $A$
@Yola It's the one-dimensional sphere in $\mathbb C^2$, i.e. $\{x\in\mathbb C^2: x^\ast x=1\}$.
May
29
revised Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$.What is numerical range $A$
added 98 characters in body
May
29
comment Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$.What is numerical range $A$
Let me apologise that I've left a wrong comment to your answer before, but your current definition of numerical range is still incorrect. The OP's one is right. $W(A)$ is the set of all values of $x^\ast Ax$ for all complex unit vectors $x$, even when $A$ is real.