How this calculation works
A square standard-coordinate matrix describes the linear operator. The input B-coordinate vector is first converted to standard coordinates, A is applied, and the output is converted back using B inverse. This changes the representation, not the underlying operator.
[T]B = B⁻¹ A BA worked example
For A = diag(2,3) and B = [1 1; 1 −1], the B-coordinate operator is [5/2 −1/2; −1/2 5/2].
Good to know
Exact rational matrices up to 6 × 6. Entries are bounded to ±1,000,000, integer fraction denominators to 1,000,000 and decimal input to 6 places. Intermediate fractions have a size limit. No complex entries, symbolic expressions, measurement uncertainty or numerical-conditioning analysis. Ordered vectors are entered as matrix columns.
Reference: Ohio State University: change of basis (opens in a new tab) ↗
An operator and its coordinate matrix are different objects
A linear operator acts on vectors. A matrix represents that action relative to a chosen basis. Changing the basis changes the matrix entries while preserving the underlying action. The formula B⁻¹ A B records the three necessary stages in their correct order.
Start with a vector expressed in B coordinates. Multiplying by B gives its standard coordinates. A then applies the operator in standard coordinates, and B⁻¹ expresses the output back in the B basis.
With A = diag(2,3) and B columns (1,1), (1,−1), the new representation has diagonal entries 5/2 and off-diagonal entries −1/2. Those fractions do not mean the original stretching factors have changed; the chosen directions mix the standard axes.
A change of basis converts a single vector’s coordinate description. Re-expressing an operator needs both the input and output conversion, producing B⁻¹AB.
Verify one column at a time
Column j of the new operator matrix gives the B coordinates of A applied to basis vector j. Multiply that returned column by B and compare it with A times column j of B. Repeating the check verifies the whole representation.
Both the input and output basis are B here. A map between different spaces or with different source and target bases requires the more general form C⁻¹ A B. This page is intentionally for a square operator using one common basis.
The result is a similarity transformation. Using only B⁻¹ A would convert the output coordinates while leaving the input in standard coordinates, which answers a different question. Keep matrix order visible when working through a handwritten solution.
A few questions, answered
Must B be an eigenvector basis?
No. Any invertible basis matrix is valid. An eigenvector basis can diagonalize an operator when enough independent eigenvectors exist, but this tool does not find them.
Does this calculate A inverse?
No. It calculates B⁻¹ A B. Only the basis matrix is inverted as part of coordinate conversion.
What if A is singular?
A singular operator is supported. B must be invertible because it represents a basis.