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  "body": "\n## Definition\n\nLet $R$ be a ring and let $A_1, \\dots, A_k$ be square matrices over $R$, where $A_i$ has order $n_i$.\nThe *block diagonal matrix* $\\operatorname{diag}(A_1, \\dots, A_k)$ is the [[block matrix|0ASDFEQS6P]] of order $n_1 + \\cdots + n_k$\n$$\\begin{pmatrix} A_1 \u0026 0 \u0026 \\cdots \u0026 0 \\\\ 0 \u0026 A_2 \u0026 \\cdots \u0026 0 \\\\ \\vdots \u0026 \\vdots \u0026 \\ddots \u0026 \\vdots \\\\ 0 \u0026 0 \u0026 \\cdots \u0026 A_k \\end{pmatrix},$$\nwhose block in position $(i, i)$ is $A_i$. The other blocks are zero matrices.\n\n## Remarks\n\nThis matrix is also called the *direct sum* $A_1 \\oplus \\cdots \\oplus A_k$.\nProducts of block diagonal matrices with matching block sizes are computed block by block.\nOver a commutative ring, $\\det \\operatorname{diag}(A_1, \\dots, A_k) = \\det A_1 \\cdots \\det A_k$.\nThe [[Jordan normal form|085HQ5G5TE]] is a block diagonal matrix of [[Jordan blocks|07PH4RX0X5]].\n\n## Explanations\n\n### L1 human\n\nA block diagonal matrix has square blocks along the diagonal and zeros everywhere else.\nEach block acts on its own group of coordinates and ignores the other coordinates.\nSo we can study each block separately.\n",
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