Let A be an n ×n matrix with eigenvalues λ1, …, , and B an m ×m matrix with eigenvalues μ1, …, .

Step 1:u000ADefine the block diagonal matrix u005C( D u005C) as given:u000Au005C[ D u003D u005Cbegin{pmatrix} A u0026 0 u005Cu005C 0 u0026 B

Let A be an n ×n matrix with eigenvalues λ1, …, , and B an m...

Let $A$ be an $n \times n$ matrix with eigenvalues $\lambda_1, \ldots, \lambda_k$, and $B$ an $m \times m$ matrix with eigenvalues $\mu_1, \ldots, \mu_l$. Show that the $(m+n) \times(m+n)$ block diagonal matrix $D=\left(\begin{array}{ll}A & 0 \\ O & B\end{array}\right)$ has eigenvalues $\lambda_1, \ldots, \lambda_k, \mu_1, \ldots, \mu_l$ and no others. How are the eigenvectors related?

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Key Concepts

Eigenvectors

Eigenvectors associated with a matrix are vectors that do not change direction under the transformation defined by the matrix. For block diagonal matrices, an eigenvector of a specific block can be extended to the larger space by placing the eigenvector in the corresponding subspace and filling other positions with zeros. This reflects the direct sum structure of the overall space, where each block contributes its own set of eigenvectors.

Block Diagonal Matrices

A block diagonal matrix is one that is composed of smaller square matrices arranged along its diagonal with all off-diagonal blocks being zero. This structure allows one to analyze each block independently when determining properties like eigenvalues and eigenvectors, since the overall matrix operates independently on the subspaces corresponding to each block.

Eigenvalues

Eigenvalues are scalars that satisfy the equation Av = ?v for a given square matrix A and nonzero vector v. In the context of block diagonal matrices, the eigenvalues of the entire matrix are exactly the eigenvalues of the individual blocks, which are simply combined together. This results from the fact that the blocks act independently on their respective subspaces.

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