Similar Matrix

Definition

• $A$
  • $n \times n$ matrix
• $B$
  • $n \times n$ matrix

$B$ is said to be similar to an $n \times n$ matrix $A$ if there exists an $n \times n$ nonsingular matrix $C$ such that

\[B = C^{-1}AC\]

■

Theorem1

• $A$
  • $n \times n$ matrix
• $C$
  • $n \times n$ nonsingular matrix

Then

• (1) \(\mathrm{rank}(C^{-1}AC) = \mathrm{rank}(A)\),
• (2) \(\mathrm{det}(C^{-1}AC) = \mathrm{det}(A)\),
• (3) \(\mathrm{tr}(C^{-1}AC) = \mathrm{tr}(A)\),
• (4) \(C^{\mathrm{T}}AC\) has same characteristic polynomial,
• (5)
• (6)

proof.

From Theorem 12.3.1 in Harville, David A. Matrix algebra from a statistician’s perspective. Vol. 1. New York: Springer, 1997.

$\Box$

Reference

• Harville, David A. Matrix algebra from a statistician’s perspective. Vol. 1. New York: Springer, 1997.