# Questions tagged [projection-matrices]

This tag is for questions relating to projection matrix, which is an square matrix that gives a vector space projection from to a subspace.

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### Reading off probabilities for measurement outcome rather than using projection operator?

Let $\alpha_{0}$ = $\alpha_{1}$ = $\frac{1}{\sqrt{2}}$. Suppose the state vector $| \psi \rangle = \alpha_{0}| \psi_{0} \rangle + \alpha_{1} |\psi_{1}\rangle$ describes a quantum mechanical system ...
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### Finding a relation between $P$ and $P^2$

I encountered a challenging problem on a mock test today: Let $x$ be an $n \times 1$ matrix. We define $$P = -(x^Tx)^{-1}\cdot(xx^T)$$ Now there were $4$ options, something like $P^2 - P = O$ where $O$...
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### $A: R^{n} \rightarrow R^{n}$, $A^{3}$ - projection. Eigenvalues and diag matrix in any basis.

Here is task: $$A: R^{n} \rightarrow R^{n}$$ $A^{3}$ - projection. (a) - What eigenvalues this linear operator have? (b) - Is it true that A will have a diagonal matrix in some basis $R^{n}$? =========...
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### Projection operator (redundant?) definition

In his Quantum Mechanics, Ballentine says that “in general, a self-adjoint operator which obeys $\rho^2=\rho$ is a projection operator”. I'm not sure I follow the need for the self-adjoint caveat? I ...
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### Geometrically, why do we need independent columns in a matrix $A$ when computing the projection matrix onto the column space of $A$?

Consider an attempt to find the line $f(t)=C+Gt+Ht$ that best approximates a set of points using least squares. This is a contrived example to try to explain what exactly goes wrong when we have non-...
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I need help with b. Lets call the column vectors of the transformation matrix $w_1, w_2, w_3$. I can already see that $w_3 = \begin{bmatrix} 1\\ 2\\ 2 \end{bmatrix}$ or simply the norm. But I am ...
I am reading through this paper by Stewart on oblique projectors, i.e. matrices $P \in \mathbb{C}^{n \times n}$ where $P^2 = P$. He describes a canonical form for projectors as follows. Let $P$ be be ...