Construct a block diagonal matrix with symbolic entries:

Show the elements:

Get the determinant:

Normal can convert a BlockDiagonalMatrix to its ordinary representation:

Make a block diagonal matrix from a full matrix:

Extract the diagonal blocks:

BlockDiagonalMatrix objects include properties that give information about the matrix:

The "Blocks" property gives a list of the diagonal blocks:

The "RowPermutation" property encodes row permutations done to the original matrix:

The "ColumnPermutation" property encodes column permutations done to the original matrix:

The "Summary" property gives a brief summary of information about the array:

The "StructuredAlgorithms" property lists the functions that use the structure of the representation:

Structured algorithms are typically faster:

Compute the determinant:

Solve a linear system:

Compute the eigenvalues:

Compute the singular values:

When appropriate, structured algorithms return another BlockDiagonalMatrix object:

Inverting bd gives another block diagonal matrix:

Transposing bd gives another block diagonal matrix:

Multiplying bd by its transpose gives another block diagonal matrix:

Elements in BlockDiagonalMatrix are coerced to the precision of the nonzero elements of the blocks.

Exact matrix:

Machine-number matrix:

Arbitrary-precision number matrix:

Construct a block diagonal matrix from a SparseArray:

Show the elements:

The core-nilpotent decomposition of a matrix:

Assemble a block diagonal matrix from the core and nilpotent parts:

Verify the decomposition:

The Jordan decomposition of a matrix:

Split the Jordan blocks:

The Kronecker product of a diagonal matrix and a general square matrix is a block diagonal matrix:

A disconnected graph:

Disconnected graphs have adjacency matrices that can be represented as block diagonal matrices:

Extract the blocks to get the connected components:

This is equivalent to using ConnectedGraphComponents:

The efficiency of the fast Fourier transform (FFT) relies on being able to form a larger Fourier matrix from two smaller ones. Generate two small Fourier matrices of sizes p and q:

The Fourier matrix of size p q can be expressed as a product of four simpler matrices:

Show that the resulting matrix is equivalent to the result of FourierMatrix:

The discrete Fourier transform of a vector can be computed by successively multiplying the factors of the Fourier matrix to the vector:

The result is equivalent to applying Fourier to the vector:

Use SparseArray[BlockDiagonalMatrix[…]] to get a representation as a SparseArray:

Diagonal matrices are treated as block diagonal matrices with 1×1 blocks:

BlockDiagonalMatrix objects with BlockDiagonalMatrix blocks are flattened to a single BlockDiagonalMatrix:

If a given matrix cannot be split into diagonal blocks or cannot be transformed into a form with diagonal blocks, BlockDiagonalMatrix returns the matrix itself: