Finite matrices: dimensions, indexing, and transpose

Mathematical statement

An m×n real matrix M is a table of real entries indexed by row i and column j, with positive integers m,n. External indices in this pulse start at one: M(i,j) is the entry in row i and column j. Matrices of the same dimensions add entrywise. The transpose Mᵀ has dimensions n×m and places M(i,j) at Mᵀ(j,i).

Representative formula (exact TeX)

M(i,j)

Assumptions

• External row and column indices are one-based, distinct from zero-based program-array indices.
• Each exercise dimension is from one through five, with nonempty equal-length rows and safe-integer entries. Zero matrices are valid, but zero-row or zero-column dimensions are outside this pulse.
• Addition requires equal dimensions. Transpose is neither reversing rows nor taking an inverse matrix.

Symbol scope

• M — specified finite real matrix
• i — one-based row index
• j — one-based column index

Intuition, definition, and misconceptions

Matrix dimensions, an individual entry, and the whole matrix are different outputs. Rearranging the same entries may produce a different matrix. Dimension exercises vary the actual row and column counts, not irrelevant entries.

Both entry indices must lie within the declared dimensions. An entry-question identity retains only dimensions, indices, and the selected entry; changing other entries is not substantive diversity.

Entrywise addition requires equal dimensions. Transpose exchanges rows and columns: a single row becomes a single column, and only square matrices retain their dimensions. Matrix multiplication is not entrywise multiplication.

Common errors include swapping rows and columns, mixing zero-based and one-based indices, or treating transpose as reversal or inversion.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Arithmetic operations — Add entries in corresponding positions exactly.
• Real number — Specify the field containing the entries.

Verification boundary

Finite integer arrays can be recomputed completely; no new formal Evidence for general matrix representations or typed-index bridges is attached.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bridge the pinned Matrix and Fin-index APIs and transpose to the external one-based convention, retaining positive dimensions.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.