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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