7
votes
If $A + tB$ is similar to $A$ for infinitely many values of $t$, where $A$ is diagonalizable, is $B$ necessarily equal to $0$?
For the non-diagonalizable case, take
$$
A = \begin{bmatrix}
1 & 1 \\0 & 1
\end{bmatrix},
\qquad
B = \begin{bmatrix}
0 & 1 \\0 & 0
\end{bmatrix}.
$$
For all $t$ except $t=-1$, we have $...
- 98.5k
2
votes
Accepted
How unique is thin/reduced QR decomposition without $R_{ii}>0$ condition?
The sticking point in my proof is the assumption that the 𝑅 matrices are invertible (or simply that they have nonzero diagonals).
Notice for any choice of $k$
$0\lt \det\big(A^T A\big) = \det\big(...
- 7,768
1
vote
Does $X, X^t$ have same operator norm.
\begin{align}
\left\|X\right\| &= \max_{\left\|h\right\|=1} \left\|Xh\right\|\\
&= \max_{\left\|h\right\|=1} \left\|\overline X^T h\right\|\\
&= \max_{\left\|\overline h\right\|=1} \left\|\...
- 5,603
1
vote
Accepted
When does a real symmetric matrix have $LDL^{T}$ decomposition? And when is the $LDL^{T}$ decomposition unique?
Let us answer the questions one by one:
Point1: The factorization $LDU$ relies on the factorization $LU$ and then we use $D$ to make $U$ as an upper triangular matrix with $1$'s in the main diagonal. ...
- 1,748
1
vote
Accepted
To compute Hermite Normal Form H and U
To undestand how the process works, it is best illustrated with a non-squared matrix. As square matrices are easier to work with, it doesn't show the full extend of the process.
Process:
The intuitive ...
- 114
1
vote
Gram matrix of QR decomposition
The projection $\hat x$ of $x$ onto the image of $A$ is
$$
\hat x = A(A^T A)^{-1} A^T x. \tag{1}
$$
so $y = x - 2 A(A^T A)^{-1} A^T x $.
This is a well known formula in the topic "linear ...
- 5,848
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