Types of Matrices

Linear Algebra (Matrices) · Axiom Academy

The special shapes and patterns — square, diagonal, identity, triangular, symmetric, and the transpose — that recur all through linear algebra. A square matrix has the same number of rows and columns. Squareness is what lets a matrix represent a transformation from a space back to itself — and it is the prerequisite for determinants, eigenvalues, and inverses. The thinnest matrices have a single line of entries. A row vector is one row wide; a column vector is one column tall. Every matrix is just a stack of row vectors — or, equally, a row of column vectors. The zero matrix O has every entry equal to 0 . It is the additive identity of matrix algebra: adding it changes nothing, exactly like adding 0 to an ordinary number. For any matrix A of the same size, A + O = A . Multiplying any matrix by a zero matrix yields a zero matrix. A diagonal matrix is square with every off-diagonal entry equal to 0 — only the main diagonal may be non-zero. That sparse pattern makes powers, inverses, and eigenvalues almost trivial to read off. D^k just raises each diagonal entry to the power k . The diagonal entries are exactly the eigenvalues. The identity matrix I_n is the diagonal matrix whose diagonal entries are all 1 . It is the multiplicative identity: multiplying by it leaves any compatible matrix unchanged, just like multiplying a number by 1 .

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