Notes in Matrix Computations

Nan Jiang –

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This is just a collections of specific terms in linear algebra and matrix computations. Instead of providing the calculation function, it just provides a direct view to explain these terms.

  • span: The “span” of \(\vec v\) and \(\vec w\) is the set of all the linear combinations.
\[a\vec v + v\vec w\]

where \(a\) and \(b\) vary over all real numbers.

  • linear transformation: compositions of “rotation” and “shear” operation. Applying one transformation after another.

  • Determinant: to measure the fact of by which the given region increase or decreases.
    • \(det(M)=0\) mean it squash the matrix into lower dimension.
    • negative determinant: flip over the orientation of your axes.
  • Duality:
  • dot product \(\leftrightarrow\) matrix-vector product.
\[\begin{bmatrix} u_x &u_y\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=u_x x+u_y y\] \[\begin{bmatrix} u_x \\ u_y\end{bmatrix}\cdot \begin{bmatrix}x\\ y\end{bmatrix}=u_x x+u_y y\]
  • cross product:

    Norm

Egienvalue, Egienvector

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References

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