Leading minors of a matrix

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You may have read before in this course, that a minor of $A$, is the matrix $A$, after we removed a line and a column. Well, leading minors (also called principal minors) are easy to find too, at least as easy as calculating a bunch of determinants.

\[ \begin{split}A= \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}\end{split} \]

Then a leading minor would be evaluated as $\Delta_{i}=det(A_{i})$ giving us something like

  • \(\Delta_{1}=a_{11}\)
  • \(\Delta_{2}=det A_{2}= \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix} = a_{11} * a_{22} - a_{12} * a_{21}\)
  • ...
  • \(\Delta_{n}=det(A)\)

We are calling $k$, for the leading minor $\Delta_k$, the order of the minor. This is the number of lines and columns we are picking, starting from 1, from the matrix $A$.