A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues

Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra

Problem 63

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 $B^2=A$.

LoadingAdd to solve later

Sponsored Links


Hint.

Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix.

Proof.

Since $A$ is a real symmetric matrix, it can be diagonalizable by a real orthogonal matrix $P$.
Thus we have
\[P^{-1}AP=D,\] where $D$ is the diagonal matrix whose diagonal entries are the eigenvalues of $A$.

The matrix $D$ is real since all the eigenvalues are real.
(Note that this is always true for a real symmetric matrix.)


If we have $B^2=A$ for some $B$, then we have
\[(P^{-1}BP)(P^{-1}BP)=P^{-1}AP=D.\]

From this, we see that we can choose $B=PD’P^{-1}$ where $D’$ is the diagonal matrix whose $i$-th diagonal entry is the square root of $i$-th diagonal entries of $D$.
The matrix $D’$ is real since the eigenvalues are non-negative by the assumption, hence the square roots of them are still real.

Since $P$ is also real, the matrix $B$ is real and satisfies
\[B^2=(PD’P^{-1})(PD’P^{-1})=PD’^2P^{-1}=PDP^{-1}=A.\]

Therefore the matrix $B$ satisfies the conditions of the problem.

Related Question.

This problems is a generalization of a liner algebra exam problem of Princeton University.

See part (b) of problem A square root matrix of a symmetric matrix.

Another related problem is the following.

Problem.
Prove that a positive definite matrix has a unique positive definite square root.

For a solution of this problem, see the post
A Positive Definite Matrix Has a Unique Positive Definite Square Root


LoadingAdd to solve later

Sponsored Links

More from my site

  • Diagonalizable by an Orthogonal Matrix Implies a Symmetric MatrixDiagonalizable 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 […]
  • A Square Root Matrix of a Symmetric MatrixA Square Root Matrix of a Symmetric Matrix Answer the following two questions with justification. (a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix. (b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]
  • Quiz 13 (Part 1) Diagonalize a MatrixQuiz 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 […]
  • Diagonalizable Matrix with Eigenvalue 1, -1Diagonalizable Matrix with Eigenvalue 1, -1 Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues. Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix. (Stanford University Linear Algebra Exam) See below for a generalized problem. Hint. Diagonalize the […]
  • Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?Square Root of an Upper Triangular Matrix. How Many Square Roots Exist? Find a square root of the matrix \[A=\begin{bmatrix} 1 & 3 & -3 \\ 0 &4 &5 \\ 0 & 0 & 9 \end{bmatrix}.\] How many square roots does this matrix have? (University of California, Berkeley Qualifying Exam)   Proof. We will find all matrices $B$ such that […]
  • Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is EvenEigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$. Then prove the following statements. (a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number. (b) The rank of $A$ is even.   Proof. (a) Each […]
  • Symmetric Matrices and the Product of Two MatricesSymmetric 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 […]
  • Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible.Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible. Let \[A=\begin{bmatrix} 1 & 3 & 3 \\ -3 &-5 &-3 \\ 3 & 3 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 4 & 3 \\ -4 &-6 &-3 \\ 3 & 3 & 1 \end{bmatrix}.\] For this problem, you may use the fact that both matrices have the same characteristic […]

You may also like...

1 Response

  1. 08/09/2016

    […] problem A square root matrix of a symmetric matrix with non-negative eigenvalues for a more general question than part […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Linear Transformation problems and solutions
If the Images of Vectors are Linearly Independent, then They Are Linearly Independent

Let $T: \R^n \to \R^m$ be a linear transformation. Suppose that $S=\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a subset of $\R^n$ such...

Close