Determinants
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Important Properties
Determinant is how the volume of the transformation changes
- determinant is just the product of eigenvalues (provable from the characteristic polynomial
- determinant is multilinear, so if you scale one column / row the whole determinant scales. Therefore, where is the dimension of
- If diagonal or upper or lower-triangular, then the determinant is just the product of the diagonal of the mattrix
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- If the matrix is not full rank, then and conversely too
- Switching rows or columns of the matrix flips the sign of the determinant
- for real matrix
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- where is eigenvalue
Different interpretations
Recursive definition of the determinant
If we let be the matrix with the ith row and the jth column removed (known as the minor
), then we have
This fact is provable but it’s a little ugly so let’s just leave it be.
We can also define the adjoint
recursively, as follows:
From this, we define the inverse
Permutation interpretation
A determinant is just the sum of all possible product permutations in a matrix. We define a permutation as the ways of scrambling . more precisesly, we generate a product for every permutation and add it up in alternating fashion to make the determinant
For more information, refer to the math 113 notes
Geometric interpretation
the determinant is the signed volume of a parallelepiped spanned by the column vectors of
Multilinear function interpretation
A determinant is a multilinaer function (i.e. it takes in the columns of a matrix and it is linear WRT to each of the columns). It is also alternating
. it is also the only such function that exists!
Properties
Multilinearity
Determinants are linear, and multilinear
Alternating
determinants are alternating
Transposes, products, and inverses
determinants are not affected by transposes (this hints at a deeper row-column duality)
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determinants are multiplicative:
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A matrix is invertible if and only if and if
Determinants of a diagonal, or U or L matrix is just the product of the diagonals.
Determinants of a block diagonal matrix is the product of the determinant of each diagonal block matrix
Determinants of an orthogonal matrix is , which hint at how these classes of matrices express rigid motions
Row and column operations
Taken together, we understand that adding any row to another row, or column to another column does not change the determinant of a matrix
- this is because you can split it up, and the component with the duplicate column has a determinant of zero
Determinant properties: summed up
- multiplying a single row or column in by multiplies the determinant by
- Exchanging a pair of rows or columns flips the sign of the determinant
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- If is singular, then
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Gradient of the Determinant
You can just use the summation definition
which means that
which means that