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In symmetric matrices,

  1. Product of pivots = determinant of that matrix
  2. Determinant of the matrix = Product of eigenvalues
  3. Therefore the product of eigenvalues = product of pivots.

Do any or all of the above apply to matrices that are not symmetric?

Moo
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Leo
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    2 is true for any square matrix. https://math.stackexchange.com/a/507660/169852 For the others, can you clarify what you mean by pivots? –  Aug 03 '19 at 20:39
  • Pivots: If a matrix is in row-echelon form, then the first nonzero entry of each row is called a pivot, and the columns in which pivots appear are called pivot columns. – Leo Aug 03 '19 at 22:07
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    Elementary row operations do not in general preserve eigenvalues. Moreover, in the usual definition of row-echelon form the pivots are all $1$, so their product is always $1$. If your definition of row-echelon form doesn’t require this, then the only thing that you can say about the pivots is that they’re nonzero: each row can be scaled arbitrarily. – amd Aug 03 '19 at 23:44
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    What are the pivots of the zero matrix? – amd Aug 03 '19 at 23:49
  • Pivots are all one only in RREF, and in REF they are not all one! – Leo Aug 03 '19 at 23:54
  • I should have specified that there are no row exchanges in arriving at row-echelon form. – Leo Aug 04 '19 at 02:12
  • If I multiply a row by some scalar, that changes the pivot value: So “product of pivots” is not well defined without some additional constraint: –  Aug 04 '19 at 03:44
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    I think the constraint OP should have specified is that you are not allowed to exchange rows or multiply rows by a scaler, only operation that is allowed it to add a multiple of one row to another and the resulting matrix must be upper triangular. I believe this will lead to a unique solution where all the statements made by OP are true. – A Kareem Aug 01 '21 at 05:11

1 Answers1

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I'm not sure what you mean by the "pivots"; there is no unique row-echelon form for any particular matrix and moreover Gaussian elimination (in particular, row-swapping and row-scaling) does not preserve determinants.

But you can say the following: elementary matrices that add a multiple of one row to a different row have determinant 1. It follows from multiplicity of the determinant that applying any sequence of such elementary operations (only) to any matrix (symmetric or not) leaves the determinant unchanged. If the resulting matrix is upper-triangular, the determinant of the matrix is the product of the diagonal entries.

As for property (2); as the constant term in the characteristic polynomial the determinant of a matrix is always the product of its eigenvalues, with appropriate algebraic multiplicity. Note that if the matrix is not symmetric, the determinant is the product of all eigenvalues, not just the real ones.

user7530
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