Linear Algebra

If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal?

Problem 647

Recall that a matrix $A$ is symmetric if $A^\trans = A$, where $A^\trans$ is the transpose of $A$.

Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it.

Otherwise, provide a counterexample.

Add to solve later

Proof.

This is true.

If $A$ is in reduced row echelon form, then every term below the diagonal must be 0.
That is, the entry $a_{i j} = 0$ for all $i > j$.

If $A$ is, additionally, symmetric, then for $i < j$ we also have $a_{j i} = a_{i j} = 0$.

These two facts together means that $a_{i j} = 0$ whenever $i \neq j$.

In this case, the only non-zero terms in the matrix $A$ lie on the diagonal, and so $A$ is a diagonal matrix.

Add to solve later

More from my site

Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2 (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1. (b) Find all such matrices with rank 2. Solution. (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1. First we look at the rank 1 case. […]
A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers. Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$. Hint. Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix. […]
Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix. Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
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 […]
Quiz 13 (Part 1) Diagonalize a Matrix Let \[A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]
$Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$$ Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$ Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix. Definition (Symmetric Matrix). A matrix $A$ is called symmetric if $A^{\trans}=A$. In terms of entries, an $n\times n$ matrix $A=(a_{ij})$ is […]
Questions About the Trace of a Matrix 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 […]
Normal Nilpotent Matrix is Zero Matrix A complex square ($n\times n$) matrix $A$ is called normal if \[A^* A=A A^*,\] where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$. A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero […]