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Results tagged with matrix-analysis
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user 1898
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
17
votes
Accepted
Closed form solution for $XAX^{T}=B$
$B^{-1/2}XAX^TB^{-1/2}=I$, so $B^{-1/2}XA^{1/2}=Q$ must be orthogonal. On the other hand, for any orthogonal $Q$, it is simple to verify that $X = B^{1/2}QA^{-1/2}$ solves the equation, so this is a c …
- 20.2k
14
votes
Accepted
Are Diagonally dominant Tridiagonal matrices diagonalizable?
Counterexample:
$$
\begin{bmatrix}
-1 & 1 & 0 & 0\\
0 & -1 & 1 & 0\\
0 & 0 & -2 & 2\\
0 & 0 & 2 & -2
\end{bmatrix}
$$
is defective: its eigenvalues are $-1,-1, 0, -4$ (it is block triangular, so its …
- 20.2k
14
votes
Accepted
Is every real matrix conjugate to a semi antisymmetric matrix?
Yes. Every matrix can be written as the sum of a symmetric plus an antisymmetric one: $A = \frac{A+A^T}{2}+\frac{A-A^T}{2}$. Now change basis such that the symmetric part is diagonal.
- 20.2k
12
votes
Matrix elements of exponential of tridiagonal matrices
Yes! Most methods to compute exponentials of large sparse matrices are based on computing directly $\exp(A)b$ for a given vector $b$ rather than the full matrix $\exp(A)$. Just take $b$ as a vector of …
- 20.2k
11
votes
Accepted
When the sum of positive definite matrices converges, does the sum of the norm of the associ...
You can bound $\|A_k\| \leq C(n)\max_{i,j} |(A_k)_{ij}|$ for some function of the dimension only $C(n)$, because all norms are equivalent in finite dimension. If I am not mistaken $C(n)=\sqrt{n}$, but …
- 20.2k
9
votes
Accepted
Source for roots of matrix polynomials?
As Geoff Robinson says, a Jordan form takes you quite far. Evaluating a scalar polynomial (or an analytic function) $f(x)$ at a Jordan block $J_{\lambda,t}$ of size $t$ and eigenvalue $\lambda$ gives …
- 20.2k
8
votes
Accepted
How to solve a non-homogeneous quadratic matrix equation?
(commenting about the equation with the plus sign, I hope that the correction was right).
This is one of the few quadratic matrix equations that have a closed form solution. Set $A=-H^{-1}$; then $G …
- 20.2k
8
votes
About Sylvester's determinant
The common ground of those two formulas is related to the Woodbury matrix identity. This relation is a useful statement that shows what happens to the inverse when one "updates" a matrix $A\in\mathbb{ …
- 20.2k
7
votes
Is this inequality involving the Frobenius norm right?
For a short fat matrix $G$ (more columns than rows), $\|AG\|_F \geq \sigma_{\min}(G)\|A\|_F \geq n \sigma_{\min}(G) \|A\|$, where $\sigma_{\min}(G)$ is the least singular value of $G$. This follows fr …
- 20.2k
7
votes
Accepted
upper bounds on a certain matrix norm
You can use the surprising identity $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$, and take the norms of the three factors separately.
- 20.2k
5
votes
How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$
To get a hang of the behaviour of matrix powers, you should consider powers of Jordan blocks:
$$
J_k(\lambda)^n = \begin{bmatrix}
\lambda^n & \binom{n}{1}\lambda^{n-1} & \binom{n}{2}\lambda^{n-2} & \c …
- 20.2k
5
votes
Accepted
Solving a vector of quadratic equations
Shameless advertisement to a paper of mine: http://www.sciencedirect.com/science/article/pii/S0024379511004484 Quadratic vector equations, in Linear Algebra and its Applications, volume 438, 2013. Ar …
- 20.2k
5
votes
Optimizing the condition number
I have been working recently on a similar problem, namely, maximizing the absolute value of the determinant of a chosen $m\times m$ submatrix $S$ of a $m\times N$ submatrix $V$ (think to the columns o …
- 20.2k
5
votes
Fast Upper Triangular Matrix Exponentiation
You probably want the Schur-Parlett method for computing matrix functions. It is a method to compute a generic function of a triangular matrix. Essentially, you apply the function to its diagonal elem …
- 20.2k
5
votes
Accepted
Norm numerical range
If $A=R^*R$ is the Cholesky factorization of $A$, then $$\frac{(ABv,v)}{(Av,v)} = \frac{(R^*RBv,v)}{(R^*Rv,v)} = \frac{(RBv,Rv)}{(Rv,Rv)} = \frac{(RBR^{-1}w,w)}{(w,w)}$$ for $w=Rv$, hence $W_A(B) = W( …
- 20.2k