**Linear Algebra**

# Symmetric matrices and their basic properties

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# Symmetric matrix

Matrix $\boldsymbol{A}$ is called symmetric if and only if $\boldsymbol{A}^T=\boldsymbol{A}$. This means that a matrix is symmetric if the matrix remains unchanged after flipping the rows and columns.

# Symmetric matrices

The following are examples of symmetric matrices:

# Symmetric matrices are square

If $\boldsymbol{A}$ is a symmetric matrix, then $\boldsymbol{A}$ is square.

Proof. Let $\boldsymbol{A}$ be an $m\times{n}$ matrix, which means that the shape of $\boldsymbol{A}^T$ is $n\times{m}$. By definition, if matrix $\boldsymbol{A}$ is symmetric, then $\boldsymbol{A}^T=\boldsymbol{A}$. For this equality to hold, the shapes of $\boldsymbol{A}^T$ and $\boldsymbol{A}$ must match, that is, $m=n$. This completes the proof.

# Sum of two symmetric matrices is symmetric

If $\boldsymbol{A}$ and $\boldsymbol{B}$ are symmetric matrices, then $\boldsymbol{A}+\boldsymbol{B}$ is also symmetric.

Proof. Let $\boldsymbol{A}$ and $\boldsymbol{B}$ be symmetric matrices. To show that $\boldsymbol{A}+\boldsymbol{B}$ is symmetric, we must show that the following holds:

By the propertylink of transpose, we have that:

Because $\boldsymbol{A}$ and $\boldsymbol{B}$ are symmetric, $\boldsymbol{A}^T=\boldsymbol{A}$ and $\boldsymbol{B}^T=\boldsymbol{B}$. Therefore, we end up with:

This completes the proof.

# Scalar multiples of a symmetric matrix is also symmetric

If $\boldsymbol{A}$ is a symmetric matrix, then $k\boldsymbol{A}$ is also symmetric for any scalar $k$.

Proof. Let $\boldsymbol{A}$ be a symmetric matrix and $k$ be any scalar. By the propertylink of matrices, we have that:

By definition of symmetric matrices, $k\boldsymbol{A}$ is symmetric. This completes the proof.

# If A and B are symmetric, then AB+BA is symmetric

If $\boldsymbol{A}$ and $\boldsymbol{B}$ are symmetric matrices, then $\boldsymbol{AB}+\boldsymbol{BA}$ is a symmetric matrix.

Proof. The transpose of $\boldsymbol{AB}+\boldsymbol{BA}$ is:

Here:

the first equality follows from theoremlink.

the second equality follows from theoremlink.

By definition of symmetric matrices, we conclude that $\boldsymbol{AB}+\boldsymbol{BA}$ is symmetric. This completes the proof.

# Powers of symmetric matrices is also symmetric

If $\boldsymbol{A}$ is a symmetric matrix, then $\boldsymbol{A}^n$ is symmetric where $n$ is any positive integer.

Proof. The transpose of $\boldsymbol{A}^n$ is:

Here, we used theoremlink for the second equality. $\boldsymbol{A}^n$ is thus symmetric by definition. This completes the proof.

# Symmetric matrix is invertible if and only if its inverse is symmetric

Let $\boldsymbol{A}$ be a symmetric matrix. $\boldsymbol{A}$ is invertible if and only if $\boldsymbol{A}^{-1}$ is symmetric.

Proof. By propertylink of invertible matrices:

By definition, $\boldsymbol{A}^{-1}$ is therefore symmetric. The backward proposition also holds because every step is based on equality. This completes the proof.

# Product of symmetric matrices is commutative if and only if the product is symmetric

Let $\boldsymbol{A}$ and $\boldsymbol{B}$ be two symmetric matrices. $\boldsymbol{AB}=\boldsymbol{BA}$ if and only if $\boldsymbol{AB}$ is symmetric.

Proof. We will first prove the forward proposition. Assume $\boldsymbol{AB}=\boldsymbol{BA}$. The transpose of $\boldsymbol{AB}$ is:

By definition, $\boldsymbol{AB}$ is therefore symmetric.

Next we will prove the converse. Assume $\boldsymbol{AB}$ is symmetric. $\boldsymbol{AB}$ is:

This completes the proof.

# Symmetric matrices and dot products

An $n\times{n}$ matrix $\boldsymbol{A}$ is symmetric if and only if the following is true:

Where $\boldsymbol{x}$ and $\boldsymbol{y}$ are any vectors in $\mathbb{R}^n$.

Proof. We prove the forward proposition first. Suppose $\boldsymbol{A}$ is an $n\times{n}$ symmetric matrix. For any vectors $\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^n$, we have that:

Note that the final step follows by theoremlink.

Now, let's prove the converse:

Therefore $\boldsymbol{A}=\boldsymbol{A}^T$, which means $\boldsymbol{A}$ is symmetric. This completes the proof.

# Product of a matrix and its transpose is symmetric

If $\boldsymbol{A}$ is an $m\times{n}$ matrix, then $\boldsymbol{A}^T\boldsymbol{A}$ and $\boldsymbol{AA}^T$ are symmetric.

Proof. If $\boldsymbol{A}$ is an $m\times{n}$ matrix, then the shape of $\boldsymbol{A}^T$ is $n\times{m}$. Therefore, the shape of $\boldsymbol{A}^T\boldsymbol{A}$ is $m\times{m}$, which means $\boldsymbol{A}^T\boldsymbol{A}$ is square. Next, $\boldsymbol{A}^T\boldsymbol{A}$ can be written as:

Here, the first equality holds by propertylink of transpose. Since the transpose of $\boldsymbol{A}^T\boldsymbol{A}$ is itself, $\boldsymbol{A}^T\boldsymbol{A}$ is symmetric by definitionlink. Similarly, we can easily show that $\boldsymbol{AA}^T$ is symmetric:

By definitionlink, $\boldsymbol{AA}^T$ is symmetric. This completes the proof.