Determinant

The determinant is a function that maps square matrices to real (or complex) numbers. It can be defined by its properties. The determinant is the unique function defined on square $n\times n$ matrices that has the four following properties.

1. The determinant of the identity matrix is 1.
2. The exchange of two columns multiplies the determinant by $-1$.
3. Multiplying a column by a scalar multiplies the determinant by this scalar.
4. Adding to a column a multiple of another column does not change the determinant.

Definition using the Leibniz formula

There are multiple other definitions. One of them uses the Leibniz formula:

$$\det (A)=\sum _{\sigma \in S_n}\text{sgn}(\sigma )a_{1,\sigma (1)}\dots a_{n,\sigma (n)}$$

$S_n$ is the set of all permutations of the set $\{1,2,\dots ,n\}$. The signature $\text{sgn}(\sigma )$ is $+1$ if the permutation can be obtained with an even number of transpositions (i.e., exchanges of two entries); otherwise, it is $-1$.

Properties

• $\det (A)={\prod }_{i=1}^n{a}_{ii}$ if $A$ is a triangular matrix.
• An alternating form: when two columns are identical, the determinant is 0; if you switch two columns, the determinant changes sign.
• $\det (\alpha A)={\alpha }^n\det (A)$
• $n$-linear function: if we fix all the columns of A except column $i$, det($A$) is a linear function of $a_i$.
• $A$ singular if and only if det($A$) = 0.
• $\det (AB)=\det (A)\det (B)$
• $\det (A^{-1})=\det (A{)}^{-1}$
• $\det (A)=\det (A^T)$

Characteristic polynomial

$$p_A(z)=\det (zI-A)$$

The polynomial $p_A(z)$ has degree $n$. Its $n$ complex roots are the eigenvalues of $A$.

The multiplicity of an eigenvalue is the number of occurrences of the corresponding root in the complete factorization of the characteristic polynomial.

Eigenvalues

The determinant is equal to the product of the eigenvalues:

$$\det (A)=\prod _{i=1}^n{\lambda }_i$$

In this product, eigenvalues are repeated according to their multiplicity.

Geometric interpretation

For any square matrix A, we consider its columns $a_i$ and the $n$-dimensional parallelepiped formed by the vectors $a_i$. The determinant of A is defined as the signed volume of this parallelepiped.

The computational cost of computing the determinant is $O(n^3)$.

Eigenvalues, Orthogonal matrix and projector