A Positive Definite Matrix Has a Unique Positive Definite Square Root
Problem 514
Prove that a positive definite matrix has a unique positive definite square root.
Add to solve later Tagged: square root matrix
Add to solve later Find All the Square Roots of a Given 2 by 2 Matrix
Problem 513
Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$.
Find all the square roots of the matrix
\[A=\begin{bmatrix}
2 & 2\\
2& 2
\end{bmatrix}.\]
Add to solve later No/Infinitely Many Square Roots of 2 by 2 Matrices
Problem 512
(a) Prove that the matrix $A=\begin{bmatrix}
0 & 1\\
0& 0
\end{bmatrix}$ does not have a square root.
Namely, show that there is no complex matrix $B$ such that $B^2=A$.
(b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root matrices.
Add to solve later Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?
Problem 133
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)
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