Trace and Determinant of a Matrix

Question

Define and distinguish between the trace and determinant of a matrix. What do they tell us about the matrix?

Accepted Answer

Trace
The trace of a square matrix is the sum of its diagonal elements. Symbolically, for a matrix $A$ with elements $a_{ij}$, the trace is given by:
\begin{equation} trace(A) = \Sigma_i a_{ii} \end{equation}
The trace is a linear operation, which means that the trace of a sum is the sum of the traces:
\begin{equation} trace(A+B) = trace(A) + trace(B) \end{equation}
 	Determinant
The determinant is a scalar value derived from a square matrix. It can be thought of as a volume scaling factor. If the matrix describes a linear transformation, the determinant describes how volumes change under this transformation. If the determinant is 0, the matrix is singular (non-invertible).

The key differences and insights:

 	The trace is related to the sum of the eigenvalues of a matrix, whereas the determinant is related to the product of the eigenvalues.
 	The trace provides a measure of the overall "size" or "magnitude" of a matrix, but doesn't capture the full geometric scaling effects.
 	The determinant gives insight into the invertibility of the matrix and the volume scaling effects of the matrix as a linear transformation.

In terms of implications:

 	A determinant of 0 implies that the matrix doesn't have full rank, meaning there's linear dependence among its rows or columns.
 	The trace can give a quick sense of the matrix's size but doesn't provide detailed geometric insights like orientation or shear.

Title	Category	Subcategory	Difficulty	Status
Cramer's Rule	Linear Algebra	Theorem	Easy	Unlock
Frobenius Norm vs. Spectral Norm	Linear Algebra	Theorem	Hard	Unlock
Gram-Schmidt Process	Linear Algebra	Theorem	Easy	Unlock
Gram-Schmidt Process for Dependent Sets	Linear Algebra	Theorem	Medium	Unlock
Inverse of a Matrix	Linear Algebra	Theorem	Easy	Unlock
Ordinary Least Squares (OLD) Method	Linear Algebra	Theorem	Easy	Unlock
Rank	Linear Algebra	Theorem	Easy	Unlock
Relevance of Singular Value Decomposition	Linear Algebra	Theorem	Medium	Example

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