# GLKMatrix4 ## Overview The [`GLKMatrix4`](/documentation/GLKit/GLKMatrix4) type defines a `4x4` floating-point matrix as well as many mathematical operations commonly used to manipulate matrices. 3D graphics programming uses matrices extensively as a convenient way to represent a graphical transformation between different coordinate systems. These transformations include translation, scaling, rotation and projection; through matrix multiplication, multiple transformations can be concatenated into a single transformation matrix. The functions that manipulate [`GLKMatrix4`](/documentation/GLKit/GLKMatrix4) structures treat the inputs as immutable, instead returning a new matrix that represent the results of the operation. ## Topics ### Creating Matrices [`GLKMatrix4Make`](/documentation/GLKit/GLKMatrix4Make(_:_:_:_:_:_:_:_:_:_:_:_:_:_:_:_:)) Returns a `4x4` matrix created from individual component values. [`GLKMatrix4MakeAndTranspose`](/documentation/GLKit/GLKMatrix4MakeAndTranspose(_:_:_:_:_:_:_:_:_:_:_:_:_:_:_:_:)) Returns a `4x4` transposed matrix created from individual component values. [`GLKMatrix4MakeWithArray`](/documentation/GLKit/GLKMatrix4MakeWithArray(_:)) Returns a `4x4` matrix created from an array of component values. [`GLKMatrix4MakeWithArrayAndTranspose`](/documentation/GLKit/GLKMatrix4MakeWithArrayAndTranspose(_:)) Returns a `4x4` transposed matrix created from an array of component values. [`GLKMatrix4MakeWithColumns`](/documentation/GLKit/GLKMatrix4MakeWithColumns(_:_:_:_:)) Returns a `4x4` matrix created from four column vectors. [`GLKMatrix4MakeWithRows`](/documentation/GLKit/GLKMatrix4MakeWithRows(_:_:_:_:)) Returns a `4x4` matrix created from four row vectors. [`GLKMatrix4MakeRotation`](/documentation/GLKit/GLKMatrix4MakeRotation(_:_:_:_:)) Returns a `4x4` matrix that performs a rotation around an arbitrary vector. [`GLKMatrix4MakeXRotation`](/documentation/GLKit/GLKMatrix4MakeXRotation(_:)) Returns a `4x4` matrix that performs a rotation around the positive x-axis. [`GLKMatrix4MakeYRotation`](/documentation/GLKit/GLKMatrix4MakeYRotation(_:)) Returns a `4x4` matrix that performs a rotation around the positive y-axis. [`GLKMatrix4MakeZRotation`](/documentation/GLKit/GLKMatrix4MakeZRotation(_:)) Returns a `4x4` matrix that performs a rotation around the positive z-axis. [`GLKMatrix4MakeWithQuaternion`](/documentation/GLKit/GLKMatrix4MakeWithQuaternion(_:)) Returns a `4x4` matrix that performs a rotation based on a quaternion. [`GLKMatrix4MakeScale`](/documentation/GLKit/GLKMatrix4MakeScale(_:_:_:)) Returns a `4x4` matrix that performs a scaling transformation. [`GLKMatrix4MakeTranslation`](/documentation/GLKit/GLKMatrix4MakeTranslation(_:_:_:)) Returns a `4x4` matrix that performs a translation. [`GLKMatrix4MakeLookAt`](/documentation/GLKit/GLKMatrix4MakeLookAt(_:_:_:_:_:_:_:_:_:)) Returns a `4x4` matrix that transforms world coordinates to eye coordinates. [`GLKMatrix4MakeOrtho`](/documentation/GLKit/GLKMatrix4MakeOrtho(_:_:_:_:_:_:)) Returns a `4x4` orthographic projection matrix. [`GLKMatrix4MakePerspective`](/documentation/GLKit/GLKMatrix4MakePerspective(_:_:_:_:)) Returns a `4x4` perspective projection matrix. [`GLKMatrix4MakeFrustum`](/documentation/GLKit/GLKMatrix4MakeFrustum(_:_:_:_:_:_:)) Returns a `4x4` perspective projection matrix. ### Working With Parts of a Matrix [`GLKMatrix4GetMatrix2`](/documentation/GLKit/GLKMatrix4GetMatrix2(_:)) Returns the upper-left `2x2` section of a `4x4` matrix. [`GLKMatrix4GetMatrix3`](/documentation/GLKit/GLKMatrix4GetMatrix3(_:)) Returns the upper-left `3x3` section of a `4x4` matrix. [`GLKMatrix4GetColumn`](/documentation/GLKit/GLKMatrix4GetColumn(_:_:)) Retrieves a column from a `4x4` matrix. [`GLKMatrix4GetRow`](/documentation/GLKit/GLKMatrix4GetRow(_:_:)) Retrieves a row from a `4x4` matrix. [`GLKMatrix4SetColumn`](/documentation/GLKit/GLKMatrix4SetColumn(_:_:_:)) Returns a new `4x4` matrix with one column replaced by a new vector. [`GLKMatrix4SetRow`](/documentation/GLKit/GLKMatrix4SetRow(_:_:_:)) Returns a new `4x4` matrix with one row replaced by a new vector. ### Performing Mathematical Operations on Matrices [`GLKMatrix4Invert`](/documentation/GLKit/GLKMatrix4Invert(_:_:)) Returns the inverse of a matrix. [`GLKMatrix4Transpose`](/documentation/GLKit/GLKMatrix4Transpose(_:)) Returns the transpose of a matrix. [`GLKMatrix4InvertAndTranspose`](/documentation/GLKit/GLKMatrix4InvertAndTranspose(_:_:)) Returns the inverse transpose of a matrix. [`GLKMatrix4Multiply`](/documentation/GLKit/GLKMatrix4Multiply(_:_:)) Returns the product of two matrices. [`GLKMatrix4Rotate`](/documentation/GLKit/GLKMatrix4Rotate(_:_:_:_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a rotation around a vector. [`GLKMatrix4RotateWithVector3`](/documentation/GLKit/GLKMatrix4RotateWithVector3(_:_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a rotation around a vector. [`GLKMatrix4RotateWithVector4`](/documentation/GLKit/GLKMatrix4RotateWithVector4(_:_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a rotation around a vector. [`GLKMatrix4RotateX`](/documentation/GLKit/GLKMatrix4RotateX(_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a rotation around the x-axis. [`GLKMatrix4RotateY`](/documentation/GLKit/GLKMatrix4RotateY(_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a rotation around the y-axis. [`GLKMatrix4RotateZ`](/documentation/GLKit/GLKMatrix4RotateZ(_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a rotation around the z-axis. [`GLKMatrix4Scale`](/documentation/GLKit/GLKMatrix4Scale(_:_:_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a scaling transform. [`GLKMatrix4ScaleWithVector3`](/documentation/GLKit/GLKMatrix4ScaleWithVector3(_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a scaling transform defined by a vector. [`GLKMatrix4ScaleWithVector4`](/documentation/GLKit/GLKMatrix4ScaleWithVector4(_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a scaling transform defined by a vector. [`GLKMatrix4Translate`](/documentation/GLKit/GLKMatrix4Translate(_:_:_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a translation transform. [`GLKMatrix4TranslateWithVector3`](/documentation/GLKit/GLKMatrix4TranslateWithVector3(_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a translation transform defined by a vector. [`GLKMatrix4TranslateWithVector4`](/documentation/GLKit/GLKMatrix4TranslateWithVector4(_:_:)) Returns a new `4x4` matrix created by concatenating a matrix with a translation transform defined by a vector. [`GLKMatrix4Add`](/documentation/GLKit/GLKMatrix4Add(_:_:)) Returns a new `4x4` matrix created by performing a component-wise addition of two matrices. [`GLKMatrix4Subtract`](/documentation/GLKit/GLKMatrix4Subtract(_:_:)) Returns a new `4x4` matrix created by performing a component-wise subtraction of two matrices. ### Performing Mathematical Operations on Vectors [`GLKMatrix4MultiplyVector3`](/documentation/GLKit/GLKMatrix4MultiplyVector3(_:_:)) Multiplies a `4x4` matrix by a `3`-component vector. [`GLKMatrix4MultiplyVector3Array`](/documentation/GLKit/GLKMatrix4MultiplyVector3Array(_:_:_:)) Multiplies a `4x4` matrix by an array of `3`-component vectors. [`GLKMatrix4MultiplyVector3WithTranslation`](/documentation/GLKit/GLKMatrix4MultiplyVector3WithTranslation(_:_:)) Multiplies a `4x4` matrix by a `3`-component vector, applying translation. [`GLKMatrix4MultiplyVector3ArrayWithTranslation`](/documentation/GLKit/GLKMatrix4MultiplyVector3ArrayWithTranslation(_:_:_:)) Multiplies a `4x4` matrix by an array of `3`-component vectors, applying translation. [`GLKMatrix4MultiplyVector4`](/documentation/GLKit/GLKMatrix4MultiplyVector4(_:_:)) Multiplies a `4x4` matrix by a `4`-component vector. [`GLKMatrix4MultiplyVector4Array`](/documentation/GLKit/GLKMatrix4MultiplyVector4Array(_:_:_:)) Multiplies a `4x4` matrix by an array of `4`-component vectors. [`GLKMatrix4MultiplyAndProjectVector3`](/documentation/GLKit/GLKMatrix4MultiplyAndProjectVector3(_:_:)) Multiplies a `4x4` matrix by a position vector to create a vector in homogenous coordinates, then projects the result to a `3`-component vector. [`GLKMatrix4MultiplyAndProjectVector3Array`](/documentation/GLKit/GLKMatrix4MultiplyAndProjectVector3Array(_:_:_:)) Multiplies a `4x4` matrix by an array of `3`-component vectors. Each result is projected back to `3`-component vector. ### Data Types [`GLKMatrix4`](/documentation/GLKit/GLKMatrix4) A `4x4` matrix. ### Constants [`GLKMatrix4Identity`](/documentation/GLKit/GLKMatrix4Identity) A `4x4` identity matrix. --- Copyright © 2026 Apple Inc. All rights reserved. | [Terms of Use](https://www.apple.com/legal/internet-services/terms/site.html) | [Privacy Policy](https://www.apple.com/privacy/privacy-policy)