Continuity of the determinant function

[dextercioby]

This is something I seek a proof of.

Theorem: Let $\mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}$ be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous.

My attempt. Continuity must be judged in $\mathbb{R}$, so it should eventually go down to an epsilon-delta proof. I was first thinking to identify $\mbox{Mat}_{n\times n}(\mathbb{R}) = \mathbb{R}^{n^2}$. How do I go further?

 

[lavinia]

This is something I seek a proof of.

Theorem: Let $\mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}$ be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous.

My attempt. Continuity must be judged in $\mathbb{R}$, so it should eventually go down to an epsilon-delta proof. I was first thinking to identify $\mbox{Mat}_{n\times n}(\mathbb{R}) = \mathbb{R}^{n^2}$. How do I go further?

Note that the determinant is a polynomial in the $n^2$ coordinates of $R^{n^2}$. For instance for a $R^4$ it is $xw-yz$.

 

[math771]

To prove continuity, we need to show that for any given small change in the input matrix, there is a small change in the output determinant. In other words, if we have a sequence of matrices {A_n} that converges to a matrix A, then the corresponding determinants {det(A_n)} should also converge to det(A).

Let's start by defining what we mean by convergence of matrices. We say that a sequence of matrices {A_n} converges to a matrix A if for any given ε>0, there exists an index N such that for all n>N, the distance between A_n and A (measured by any matrix norm) is less than ε. In other words, the matrices in the sequence eventually get very close to A.

Now, let's consider the sequence of determinants {det(A_n)} corresponding to the sequence of matrices {A_n}. We want to show that this sequence also converges to det(A). To do this, we will use the definition of continuity for functions.

Let ε>0 be given. We need to find a δ>0 such that if the distance between A_n and A is less than δ, then the distance between det(A_n) and det(A) is less than ε. This is the same as saying that if the matrices in our sequence are close to each other, then their determinants are also close to each other.

Since we have identified $\mbox{Mat}_{n\times n}(\mathbb{R}) = \mathbb{R}^{n^2}$, we can use the Euclidean norm on matrices to measure their distance. So, let's choose δ=ε. Now, if the distance between A_n and A is less than δ=ε, then the distance between det(A_n) and det(A) is also less than ε, since the determinant function is a continuous function on $\mathbb{R}^{n^2}$.

Therefore, we have shown that for any given ε>0, there exists a δ>0 such that if the distance between A_n and A is less than δ, then the distance between det(A_n) and det(A) is less than ε. This is the definition of continuity, and hence we have proven that the determinant function is continuous.