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2d
answered For any skew-symmetric $3×3$ matrix $A$, does there exist a symmetric $3×3$ matrix $B$ such that $AB+BA=O$
Feb
5
answered Are there singular matrices such that if we change any entry it will be non-singular?
Feb
4
answered Matrix equation implies invertibility
Feb
3
comment Why $x\in\ker A$ implies $x_i-x_j=\lambda \det A_{ij}$?
@W-t-P Suppose $j=n$ (the case $j<n$ is similar) and let $B=\pmatrix{\widetilde{A}\\ \mathbf z^T}$ (so that $\widetilde{A}$ comprises the first $n-1$ columns of $A$). Since $\mathbf x\in\ker A$, we have $\mathbf z\in\ker \widetilde{A}$. Thus $BB^T$ is the direct sum of two diagonal blocks $\widetilde{A}\widetilde{A}^T$ and $\|\mathbf z\|^2$. As $\widetilde{A}$ is a real matrix with full row rank, $\widetilde{A}\widetilde{A}^T$ is invertible. As $\|\mathbf z\|^2$ is also invertible, $BB^T$ is invertible.
Feb
2
answered Why $x\in\ker A$ implies $x_i-x_j=\lambda \det A_{ij}$?
Jan
31
comment Matrices commuting with involution like matrices
The answer is yes. Isn't this a direct consequence of the existence of real Jordan form?
Jan
31
comment Matrix with maximal rank in a family of matrices
The claim that $A$ must be singular is true only because the field $\mathbb R$ has characteristic zero. If the underlying field has an appropriate finite characteristic, the claim may not hold. E.g. over $GF(2)$ (where $1+1=0$), we have $AB-BA=A$ when $$A=\pmatrix{1&1\\ 0&1},\ B=\pmatrix{1&1\\ 1&0}.$$
Jan
31
answered Matrix with maximal rank in a family of matrices
Jan
30
comment Suppose $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$?
$A$ and $B$ must have identical kernels. So, if you consider the restrictions of them on the subspace orthogonal to their common kernel, you may assume that both $A$ and $B$ are invertible. But then this means $\|AB^{-1}x\|_2=\|x\|_2$. It is well-known that the matrix of a norm-preserving linear map on $\mathbb R^n$ must be orthogonal. Consequently (and for the original $A$ and $B$), $\|Ax\|_2=\|Bx\|_2$ for all $x$ iff $A=QB$ for some real orthogonal matrix $Q$.
Jan
27
revised What does echelon mean?
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Jan
27
answered What does echelon mean?
Jan
27
comment Find $||\cdot||_2$ norm of block matrix
@DavidLi Just google "tridiagonal Toeplitz matrix eigenvalues" and click any relevant link.
Jan
26
comment Find $||\cdot||_2$ norm of block matrix
If you are talking about the induced 2-norm (i.e. the operator norm, a.k.a. spectral norm), the following fact is useful: the eigenvalues of an $n\times n$ tridiagonal Toeplitz matrix are given by $\lambda_j = a+2\sqrt{bc}\cos\frac{j\pi}{n+1};\,j=1,2,\ldots,n$ (where the main diagonal entries are equal to $a$, and $b,c$ are the values of the entries of the other two diagonals).
Jan
26
comment Find $||\cdot||_2$ norm of block matrix
@DavidLi Dac0 is probably trying an example with $n=3$.
Jan
26
comment Compute the square norm $||\cdot||_2$ of matrix
Thanks, David. If the answer given in the other thread is incorret, it's mainly because your definition (if any) of "square norm" is unclear. Anyway, I don't see there're any fundamental differences between the questions, so I think this question should still be closed.
Jan
26
comment Find $||\cdot||_2$ norm of block matrix
That doesn't make sense. We are talking about matrix norms, not vector norms. Your matrix has $n^4$ entries. What do you mean by $\sqrt{x_1^2+\cdots+x_n^2}$?
Jan
26
comment Compute the square norm $||\cdot||_2$ of matrix
As the OP had posted a very similar question a few hours ago, I voted to close this question.
Jan
26
answered How to prove that the inequality holds for any nonzero x?
Jan
25
answered Result of matrix $A^{2016}$
Jan
24
revised X,AX have no common eigenvalues
added 238 characters in body