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 $...
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  • 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(...
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  • 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\|\...
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  • 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. ...
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1 vote
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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 ...
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  • 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 ...
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  • 5,848

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