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.
Transpose of a Matrix and Eigenvalues and Related Questions
Let $A$ be an $n \times n$ real matrix. Prove the followings.
(a) The matrix $AA^{\trans}$ is a symmetric matrix.
(b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.
(c) The matrix $AA^{\trans}$ is non-negative definite.
(An $n\times n$ […]
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Show that eigenvalues of a Hermitian matrix $A$ are real numbers.
(The Ohio State University Linear Algebra Exam Problem)
We give two proofs. These two proofs are essentially the same.
The second proof is a bit simpler and concise compared to the first one.
[…]
Symmetric Matrices and the Product of Two Matrices
Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.
(a) The product $AB$ is symmetric if and only if $AB=BA$.
(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.
Hint.
A matrix $A$ is called symmetric if $A=A^{\trans}$.
In […]
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Suppose $A$ is a positive definite symmetric $n\times n$ matrix.
(a) Prove that $A$ is invertible.
(b) Prove that $A^{-1}$ is symmetric.
(c) Prove that $A^{-1}$ is positive-definite.
(MIT, Linear Algebra Exam Problem)
Proof.
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A real symmetric $n \times n$ matrix $A$ is called positive definite if
\[\mathbf{x}^{\trans}A\mathbf{x}>0\]
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7 Problems on Skew-Symmetric Matrices
Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$.
(a) Prove that $A+B$ is skew-symmetric.
(b) Prove that $cA$ is skew-symmetric for any scalar $c$.
(c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is […]
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Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if
\[A_1^2+A_2^2+\cdots+A_m^2=\calO,\]
where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$.
Hint.
Recall that a complex matrix $A$ is Hermitian if […]