49 lines
1.4 KiB
Markdown
49 lines
1.4 KiB
Markdown
# Geometry Formulary
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## Inverse of a Matrix
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$A^{-1} = \frac{1}{det(A)} Adj(A)$
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## Adjugate Matrix
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The adjugate of a matrix $A$ is the `transpose` of the `cofactor matrix`:\
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$Adj(A) = C^{T}$
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### $(i-j)$-minor (AKA $M_{ij}$)
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$M_{ij}$ := Determinant of the matrix $B$ got by removing the
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***$i$-row*** and the ***$j$-column*** from matrix $A$
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### Cofactors
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$C$ is the matrix of `cofactors` of a matrix $A$ where all the elements $c_{ij}$
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are defined like this:\
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$
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c_{ij} = \left( -1\right)^{i + j}M_{ij}
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$
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## Eigenvalues
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By starting from the definition of `eigenvectors`:\
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$A\vec{v} = \lambda\vec{v}$
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As we can see, the vector $\vec{v}$ was unaffected by this matrix
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multiplication appart from a scaling factor $\lambda$, called `eigenvalue`
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By rewriting this formula we get:\
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$
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\left(A - \lambda I\right)\vec{v} = 0
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$
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This is solved for:\
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$\det(A- \lambda I) = 0$
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> [!NOTE]
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> If the determinant is 0, then $(A - \lambda I )$ is not invertible, so
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> we can't solve the previous equation by using the trivial solution (which
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> can't be taken into account since $\vec{v}$ is not $0$ by definition)
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## Caley-Hamilton
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Each square matrix over a `commutative ring` satisfies its own
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characteristic equation $det(\lambda I - A)$
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> [!TIP]
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> In other words, once found the characteristic equation, we can
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> substitute the ***unknown*** variable $\lambda$ with the matrix itself
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> (***known***),
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> powered to the correspondent power |