Show that if A and B are two n ×n matrices, then the matrices A B and B A have the same characte

step 1 From this exercise, we have the determinant of A times B minus lambda i sub n, and the determina

Show that if A and B are two n ×n matrices, then the...

Show that if $A$ and $B$ are two $n \times n$ matrices, then the matrices $A B$ and $B A$ have the same characteristic polynomial, and thus the same eigenvalues (matrices $A B$ and $B A$ need not be similar though; see Exercise 55 ). Hint: \[ \left[\begin{array}{cc} A B & 0 \\ B & 0 \end{array}\right]\left[\begin{array}{cc} I_{n} & A \\ 0 & I_{n} \end{array}\right]=\left[\begin{array}{cc} I_{n} & A \\ 0 & I_{n} \end{array}\right]\left[\begin{array}{cc} 0 & 0 \\ B & B A \end{array}\right] \].

Show that if $A$ and $B$ are two $n \times n$ matrices, then the matrices $A B$ and $B A$ have the same characteristic polynomial, and thus the same eigenvalues (matrices $A B$ and $B A$ need not be similar though; see Exercise 55 ). Hint:
\[
\left[\begin{array}{cc}
A B & 0 \\
B & 0
\end{array}\right]\left[\begin{array}{cc}
I_{n} & A \\
0 & I_{n}
\end{array}\right]=\left[\begin{array}{cc}
I_{n} & A \\
0 & I_{n}
\end{array}\right]\left[\begin{array}{cc}
0 & 0 \\
B & B A
\end{array}\right]
\].

Key Concepts

Block Matrix Decomposition

Block matrix decomposition involves partitioning a larger matrix into smaller submatrices (blocks), which can simplify many matrix operations, such as multiplication and inversion. This technique is particularly useful in proving relationships between matrices, as it allows one to apply transformations and similarities in a structured way. In the context of the given problem, block matrices enable a direct connection between the products AB and BA, illustrating how their characteristic polynomials and eigenvalues are related.

Matrix Similarity

Matrix similarity is a concept that defines an equivalence relation among square matrices. Two matrices are said to be similar if there is an invertible matrix that can transform one matrix into the other via conjugation. Similar matrices represent the same linear transformation under different bases, and they share important properties such as the characteristic polynomial and eigenvalues, even though they may not be identical in form.

Eigenvalues

Eigenvalues are scalars associated with a given square matrix, characterizing the factors by which the eigenvectors are scaled under the corresponding linear transformation. They are found by solving the characteristic equation, which is derived from setting the characteristic polynomial equal to zero. Eigenvalues play a crucial role in many areas of mathematics and applications, including differential equations, stability analysis, and matrix diagonalization.

Characteristic Polynomial

The characteristic polynomial of a matrix is a polynomial which is defined by the determinant of the matrix subtracted by a scalar multiple of the identity matrix. This polynomial encodes important information about the matrix, most notably its eigenvalues, as these are exactly the roots of the characteristic polynomial. It is a fundamental concept in linear algebra that reflects key properties of the linear transformation associated with the matrix.

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