Question about hollow matrix and diagonalization

AI Thread Summary
Diagonalizing a hollow matrix, defined as a matrix with zero entries along the diagonal, is possible but depends on the specific matrix. For example, the matrix with entries [[0, 1], [1, 0]] can be diagonalized, while the zero matrix [[0, 0], [0, 0]] is trivially diagonalizable. The discussion highlights that the diagonalizability of hollow matrices is not guaranteed and varies based on their structure. Understanding the characteristics of the matrix is crucial for determining diagonalizability. Overall, the ability to diagonalize hollow matrices is conditional and requires examination of individual cases.
pizzamakeren
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Homework Statement
A simple question about the topic diagonal matrix and diagonalization.
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A quick and simple question. I just realized that this has been posted in the wrong section, but ill give it a try anyway. Does anyone know if it's possible to diagonalize a hollow matrix? What i mean by a hollow matrix is a matrix with only zero entries along the diagonal.
 
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Why not? Try diagonalizing$$\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}$$and see what you get.
 
Sometimes. It depends on the matrix.
\begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix}
is obviously diagonalizable.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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