# Questions About the Trace of a Matrix

## Problem 19

Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.

(a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of $B$.

(b) Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$.

(c) Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$.

## Hint.

Review

1. the definition of the transpose of a matrix
2. the definition of matrix multiplication
3. the definition of a symmetric matrix

Then the proofs of these statement is straightforward computations.

## Proof.

### (a) Express $\tr(AB^{\trans})$ in terms of the entries of $A$ and $B$.

Here we use the following notation for an entry of a matrix: the $(i, j)$-entry of a matrix $C$ is denoted by $(C)_{i,j}$.

Then the $(i,j)$-entry of $AB^{\trans}$ is $(AB^{\trans})_{ij}=\sum_{k=1}^n a_{ik}b_{jk}$.
Thus we have
$\tr(AB^{\trans})=\sum_{l=1}^n (AB^{\trans})_{ll}=\sum_{l=1}^n \sum_{k=1}^n a_{lk}b_{lk}.$

### (b) Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$

By the formula obtained in part (a), we have
$\tr(AA^{\trans})=\sum_{l=1}^n \sum_{k=1}^n a_{lk}^2.$ This is the sum of the squares of entries of $A$.

### (c) Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$.

Since $A$ is a symmetric matrix, we have $A^{\trans}=A$.
Thus by the result of part (b), we have

$\tr(A^2)=\tr(AA^{\trans})=\sum_{l=1}^n \sum_{k=1}^n a_{lk}^2>0.$ The last sum is strictly positive since $A$ is not the zero matrix, there is a nonzero entry of $A$ (and of course the square of a real number is nonnegative).

## Comment.

The results we proved in this article can be extended to complex matrices, matrices with complex number entries.
In this case, the condition in (c) that $A$ is symmetric is replaced by the condition that $A$ is a hermitian matrix.

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